lOMoARcPSD|13483535 300 Problems In Analytic Geometry Section 1 Engineering Managment (Western Institute of Technology) StuDocu is not sponsored or endorsed by any college or university Downloaded by Michael Berco (bercomichael29@gmail.com) lOMoARcPSD|13483535 1. Find the distance between A(5,-3) and B(2,1). Answer: 5 2. Find the slope of a line, which passes through point A(5,-3) and meets y axis at 7. Answer: - 2 3. Find the equation of a line through points P1(1,3) and P2(-2,1) Answer: y = 2 7 x+ 3 3 −1 x+1 2 5. Find the equation of a line which passes through A(4,-1) and is parallel to x axis. Answer: - 1 6. Find the equation of a line which intercept y axis at -4 and also intercept x axis at 2. Answer: 2x – y = 4 7. Find the distance between A(2,-3) and the line 3x-4y+2=0. Answer: 4 8. Find the area of a triangle with vertices A(0,-3), B(5,0), C(0,3). Answer: 15 9. What is the area of a triangle with vertices P(1,1), Q(-1,2), R(2,1). 3 2 11. Find the value of m, such that D1, D2, D3 meet each other at one point. D1: x – y=1 D2: 2x+y=5 D3: (2m-5)x – my=3 Answer: 13 3 12. If D is parallel to D’, D:2x – y = 1 and D’: (a – 1)x + 2y = x – 2, then find the value of a. Answer: - 2 4. A line with equation y=mx+b passes through the points P1(-2,2) and P2(2,0). What are the values of m and b? Answer: y = Answer: 10. If the distance of A(2x-3,5) from line x=-4 is equal to 7, find the value of a. Answer: 3,4 13. Calculate the distance of point A(2,1) from the line p: X = -1 + 3t Y = 5 – 4t Line p has a parametric form of the line equation. Answer: 0.2 14. Find the equation of the circle with center (3,7) and circumference 8 π units. Answer: (x - 3)2 + (y – 7)2 = 16 15. The line passed through three points – see table: x y -6 4 -4 3 -2 2 Write line equation in the form y=mx+b Answer: y = −1 x+1 2 16. Find the distance between the points (7,-9), (-1,-9). Answer: 8 Downloaded by Michael Berco (bercomichael29@gmail.com) 17. Find parameters of the circle in the plane coordinates of center and radius x2 + (y 3)2 = 14 Answer: x1=0, y1=3, r= √ 14 18. show that the points D(2,1), E(4,0), F(5,7) are vertices of a right triangle. Answer: 0 19. If the midpoint of a segment is (6,3) and the other end point is (8,-4) what are the coordinate of the other end? Answer: (4,10) 20. Three points A(-3,-5), B(9,-10), and C(2,k). AB = AC. What is the value of k? Answer: 7,-17 21. On line p: x = 4 + t, y = 3 + 2t, t is R, find point C, which has the same distance from points A(1,2) and B(-1,0). Answer: c(2,-1) 22. A circle is tangent to the x,y axes and is passing through the point P(3,6) the center of the circle lies on the line x+ y = 0. Find the equation of the circle. Answer: (x - 15)2 + (y – 15)2 = 152 and (x - 3)2 + (y – 3)2 = 32 Situation: P(4,-2) is parallel to the line 2x – y = 3. 23. Find the equation of the line. Answer: 2x – y = 0 lOMoARcPSD|13483535 24. Find the distance between the lines. Answer: passing through the point (0,-1). Answer: 2.43x+y+1=0 and x+y+1=0 7 √5 5 Situation: The line x + 2y – 10 = 0 is tangent to a circle who’s center is at point (2,-1). √ 10−2 25. Find the tangent point. Answer: (4,3) 26. Find the second tangent point of a line with the same slope as the given line. Answer: (0,-5) 27. Given a right triangle with base at points A(5,-3) and C(6,3), vertex B is located on the axis. Find the values of point B. Answer: B(0,4) Situation: Given the circle (x – 1)2 + (y - 4)2 = 25. The center of the circle is at M. the line x + y – 6 = 0 is cutting the circle at points A and B. 28. Find the values of points A and B. Answer: A(-2,8), B(5,1) 29. Find the slope of line MA. Answer: - 32. Calculate the components of the curve x2 + 4x – 6 = 0. Answer: ± 4 3 30. Find the tangent to circle line equation at point A. Answer: 3x– 4y+38=0 31. Line equation is given by 3x + 2y + 2 = 0. Find the equations of the lines that are located 11.3 degrees from the given line and 33. Calculate the components of the curve x2 + 4xy + 3y2 – 2xz – 4yz + z2 = 0. Answer: x+y-z=0 and x+3y-z=0 34. Calculate the real values of m such that the following conic section is degenerated. x2 - 4xy + 2y2 – 5y – mx + 2 = 0. Answer: ± 3 √ 2−5 2 35. Calculate the double points of the following conic section. x2 + 4xy + 3y2 – 2xz – 4yz + z2 = 0. Answer: x=1, y=0, z=1 36. Calculate the double points of the following conic section. x2 - 2xy + y2 – 8xz – 8yz + 16z2 = 0. Answer: x+y-4z=0 37. Calculate the tangent line in point P(2,0) of the conic section x2 4xy - y2 + 2x – 4y - 8 = 0. Answer: x-2y-2z=0 38. Calculate the asymptotes of the conic section x2 - 4xy - y2 + 2x – 4y - 8 = 0. Answer: (2x-4y+2z)± ( √ 11−4 )(-4x-2y4z) = 0 Downloaded by Michael Berco (bercomichael29@gmail.com) 39. Search for the equation of the conic section with asymptotes x = 0 and x + 2y – 41 = 0 and through the point P(2,1). Answer: x2+2xy4x+74=0 40. Determine the equation of the curve such that the sum of the distances of any point of the curve from two points whose coordinates are (-3,0) and (3,0) is always equal to 8. Answer: 7x2+16y2112=0 41. A circle is tangent to the line 2x – y + 1 = 0 at the point (2,5) and the center is on the line x + y = 9. Find the equation of the circle. Answer: x+2y=12 42. Given the circle x2 + y2 = 25 and the point A(3,-4) on the circle. Find the equation of the tangent to the circle at A. Answer: y+4= 3 4 (x-3) 43. The line with equation 2x – 3y = 10 touches the circle with center M(2,3) at point A. find the equation of the circle. Answer: (x+2)2+(y4)2=52 44. Find the distance between two points A and B such that the coordinates of A and B are (5,-3) and (2,1). Answer: 5 lOMoARcPSD|13483535 45. Determine the slope of the line that passes through the point A(5,-3) and meets at y-axis at 7. Answer: -2 Answer: 3 54. What is the xintercept of the line passing through (1,4) and (4,1). Answer: -5 46. Find the distance between (4,-3) and (2,5) Answer: 10 55. Find the equation of the circle whose center is at (3,-5) and whose radius is 4. Answer: x2 + y2 – 6x + 10y + 18 = 0. 47. The segment from (1,4) to (2,-2) is extended three times its own length. Find the terminal point. Answer: (11,-20) 56. What is center of the curve x2 + y2 – 2x – 4y – 31 = 0. Answer: c(1,2) 48. Given three vertices of a triangle whose coordinates are A(1,1), B(3,-3), and C(5,-3). Find the area of the triangle. Answer: 4 sq. Units 49. The line segment connecting (x,6) and (9,y) is bisected by the point (7,3). Find the values of x and y. Answer: x=5, y=0 50. What is the equation of the line that passes through (4,0) and is parallel to the line x – y – 2 =0? Answer: x-y-4=0 51. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0. Answer: B=3 52. Find the distance from the line 4x – 3y + 5 = 0 to the point (2,1). Answer: 2 53. Find the distance between lines 3x = y – 12 = 0 and 3x + y – 4 = 0. Answer: 2.53 64. The point (3,2) is the midpoint of the segment defined by the points (h,1) and (5,k), hand k are respectively: Answer: 3 65. The distance from the point (2,1) to line 4x-3y+5=0 is. Answer:2 66. The two points on the line 2x+3y+4=0 which are at a distance 2 from the line 3x+4y-6=0 are. Situation: Given parabola y2 + 8x – 6y + 25 = 0. 57. Compute the length of the latus rectum. Answer: 8 Answer:(64,-44) and (4,-4) 58. Compute the focal length of the parabola. Answer: 2 67. Find the slope of the line whose equation is 3x+4y=8. Answer: 59. An arch 18m has the form of parabola with a vertical axis. The length of a horizontal beam placed across the arch 8m from the top is 64m. find the width of the arch at the bottom. Answer: 96 60. Find the center of the ellipse 9x2 +25 y2 – 18x – 100y = 116. Answer: c(-1,2) 61. Add the given vectors (-4,7)+(5,-9) Answer: (1,-2) 62. Find the length of the vector (2,1,4) −3 4 68. Find the distance between A (4,-3) and (-2,3). Answer: 6 69. The 2 straight lines below 4x-y+3=0 8x-2y+6=0 are. Answer: coincident 70. Where would these 2 lines intersect? X=3 and y=-2. Answer: (3,-2) Answer: 21 63. The distance betweenthe points (1,3) and (9,k) is 8. Find the value of k. 71. A horizontal line has a slope of. Answer: zero Downloaded by Michael Berco (bercomichael29@gmail.com) lOMoARcPSD|13483535 Answer: 5x-3y=0 72. The equation r=a is the polar equation of. Answer: circle 73. Given the equation 3y+2x+1=0. Determine the line perpendicular to it. 80. The distance between lines x-2y=4 and 2x4y=7 is. 88. The center of a circle is at (1,1) and one point on its circumference is (1,3). Find the other end of the diameter through (-1,-3). Answer: 0.224 Answer: (3,5) 81. Compute the yintercept of a line passing through point (5,3) and a slope of 3/4. 89. What is the equation of the line that passes thru (0,4) and is parallel to the line x-y-2=0 Answer: 9x-6y+12=0 Answer: 74. Find the equation of line passing through the points P1(-8,1) and P2(8,-1). −3 4 82. The distance between points (4,7,z) and (5,1,6) is 7.28. Find z. Answer: x+8y=0 Answer: x-y+4=0 90. When a line y= mx+b slopes downwards from left to right, the slope m is. Answer: less than zero Answer: 2 75. A line passes through point (2,2) whose line segment intercept between the coordinates axes and has a length of the square root of 5. Find the equation of the line. Answer: 2x+y-2=0 83. Find the equation of perpendicular bisector of the line passing through points (2,-5) and (-3,4) 91. If the product of the slopes of any two straight lines is negative 1, one of these lines are said to be. Answer: perpendicular Answer: 5x-9y=2 84. Compute the shortest distance from point (26,4) to the curve x216x+y2+16y+64=0. 92.The midpoint of the line segment between P1(x,y) and P2(-2,4) is Pm(2,-1). Find the coordinate of P1. Answer: (6,-6) Answer: 13.63 76. What is the xintercept of line passing through (1,4) and (4,1)? 85. Determine the point of division of the line segment from A (5,6) to B(-3,-2) that divides this line segment, starting from A, into two parts in the ratio 1:4. Answer: 5 77. Find the equation of straight line through point (3,2) and is parallel to line y=3x-2. 93.Convert θ=π/3 to Cartesian equation. Answer: y=3^1/2x 94. Find the coordinates of the point P(2,4) with respect to the translated axis with origin at (1,3). Answer: (3,4) 86. Find the radius of the sphere from the equation x2+y2+z2+8x-2y+1=0. Answer: y=3x-7 Answer: 4 87. Determine the equation of the line bisector of the smaller angle formed y the intersection of the following lines: 78. The equation of line through point (3,-2) and is perpendicular to line 2x+3y+4=0 is. Answer: 2y-3x+13=0 95. The angle of inclination of ascend of a road having 8.25% grade is___ degrees. Answer: 4.72 degrees L2=5x-12y+30=0 96. Determine B such that 3x+2y-7=0 is perpendicular to 2x-By+2=0. Answer: 9x+33y=154 Ans: 3 L1=4x+3y-24=0 79. The equation of a line passing through the origin and parallel to the line 5x3y+8=0 is. Answer: (1,1) Downloaded by Michael Berco (bercomichael29@gmail.com) lOMoARcPSD|13483535 97. Point P(x,y) movves with a distance from point (0,1) one-half of its distance from line y=4, the equation of its locus is. Ans: 4x2+3y2=12 98. A line passed through point (2,2). Find the equation of the line if the length of the line segment intercepted by the coordinate axes is the side of a square whose area is 5 square units. Ans: 2x-y-2=0 104. If the points (-2,3), (x,y) and (-3,5) is on a straight line, then the equation of the line is. 112. The length of the latus rectum of the parabola defined by the equation (x2)2-y+3=0 is. Ans: 2x+y-1=0 Ans: 1 105. Find the inclination of the line passing through (5,3) and (10,7). 113. The parabola y=x2+x+1 opens. Ans: downward Ans: 14.63 degrees 106. Find the equation of the normal to x2+y2=1 at point (2,1). 114. The parabola y=-x26x-9 opens. Ans: upward Ans: x=2y 99. Find the area of the triangle which the line 2x3y+6=0 forms with the coordinate axes. 107. The distance between the points A and B is defined by the points :A (cos A, -sin A) and B (sin A, cos A) and is equal to. Ans: 3 100. Determine the coordinates of the point which is three-fifths of the way from the point (2,-5) to the point (3,5). Ans: (-1,1) Ans: √2 108. Given two points(3,7) and (-4,7). Solve for the distance between them. Ans: 15.65 101. 4x2-256=0 is the equation of. Ans: parallel lines 109. The locus of the parabola y2-4x is at. Ans: (1,0) 102. Two vertices of a triangle are (2,4) and (2,3) the area is 2 sq. Units, the locus of the third vertex is. Ans: x-4y=-18 103. The linear distance between -4 and 17 on the number line is. Ans: 21 110. Find the equation of the directrix of the parabola y2=16x. 115. In a conic section, if the eccentricity e>1, then, the locus is a/an _____. Ans: hyperbola 116. Find the area in the first quadrant bounded by the curve y=2x-x2 and the x- axis. Ans: 4/3 117. Find the location of the locus of the parabola y2+4y-4x-8=0. Ans: (-2,-2) 118. Find the area enclosed by the curve x2+8y+16=0, the x-axis, yaxis and the line x-4=0. Ans: 10.67 sq units Ans: -4 111. Determine the equation of the directrix of the 3=2y2-4y+7. 119. Given the equation of the curve 9x2+25y2144x+200y+751=0. Find the distance between foci. Ans: 8 units Ans: x=31/24 Downloaded by Michael Berco (bercomichael29@gmail.com) lOMoARcPSD|13483535 points (1,2) and ( -2, 5). 120.An elliptical plot of garden has semi major axis of 6 meters and semi-minor axis of 4.8 meters. If these are increased by 0.15 meter each, find the increase in the area of the garden in square meters. Ans:5.16 Ans. y=x+3 126. What is the equation of the line with a slope of 0 and passing through point (4,6 ). Ans. y=6 121. In the given equation y= mx + b, m is the slope of the line and b is the yintercept. If the coordinates of two points is x1= 7.5, x2= 12.5, y2= 20.6, y1= 8.5. what is the value of “m” Ans. m= 2.42 121. Find the equation of the line having a slope 0f -2 and a y-intercept 0f -9. Ans. y= -2x-9 122. Find the gradient and yintercept of the line y2= 4x + 9. Ans. gradient= 4 ,yintercept = 11 123. What is the slope of a vertical line? 127. What is the slope of a horizontal line? Ans. slope is zero 128. Given the equations y= x2– 4x -5 and y+ x= -1. Determine the point which satisfies both equations. Ans. (4,5) Ans. (2,1) 125. What is the equation of a line passing through the Ans. α= 59.04° 133. What is the slope of the line 3x+2y+1=0. Ans. m= -3/2 134. Find the area bounded by the line 2x-y+10=0. Ans. A=25 135. Find the distance from the point (5,3) to the line 7x-4y=0. Ans. d=2.36 129. The equations 5x + 2y = 48 and 3x + 2y = 32 represents the money collected from school concert ticket sales during the two class periods. If x represents the cost for each adult ticket and y represents the cost for each student ticket, what is the cost for each adult ticket? Ans. x=8 Ans. y/0 (undefined) 124. Find the coordinates of the point of the lines y= 3x-5 and 3y+2x=7. 132. Two lines intersect a point (2,4). One line has a slope of -1 and the other line has a slope of 1/4. Find the acute angle between the lines at point (2.4). 130. Find the gradient of the line y=3. Ans. gradient = 0 131. Find the area of a triangle whose vertices are A(-3,-1), B(5,3) and C(2,-8). Ans. A= 38 Downloaded by Michael Berco (bercomichael29@gmail.com) 136. The coordinates of the vertices ABC are A(1,4), B(3,0) and C(-2,2). Find the equation of the perpendicular bisector of the side AB. Ans. x-2y+2=0 137. The sum of the coefficients of x and y in Ax+By-16=0 is 14. If the slope of the line is 8. Find the value of B. Ans. B=-2 138. Find the equation of the circle with center at (-3,8) and tangent to the line x-y+5=0. lOMoARcPSD|13483535 Ans. x2+y2+6x16y+55=0. 139. Find the distance from point (6,4) to the circle x216x+y2+16y+64=0. Ans. d=12.17 140. Find the equation of the circle which passes through the point (1,5) and touching the y-axis with its center along the xaxis. Ans. x2+y2-26x=0 141. A circle passes thru the origin and point (4,2). Find the equation of the circle if its center along the y-axis. passes through the point (2,5). What is the equation of the circle. Ans. (x-4)2+(y-3)2=40 Ans. 12 146. Two circles whose equations are (x-3)2 + (y-5)2 = 25, and (x-7)2 +(y-5)2 = 9 intersect in two points. What is the equation of the line passing through these two points? Ans. x=7 147. Find the equation of the circle which is tangent to both axes, center in second quadrant, r=2. Ans. x2+ 4y+4=0 y2+4x- Ans. x2+y2-10y=0 142. A circular rotunda passes thru three points A(0,0), B(2,2), and C(5,0). Find the radius of the rotunda. Ans. r= 2.55 143. What is the radius of the circle x2+y2-6y=0? Ans. r=3 144. Find the area of the region bounded by the 2 2 curve x +y +8x+4y61=0 Ans. A= 254.47 sq. units 145. A circle has a center at (-4,3) on the xy plane and 151. Determine the length of the latus rectum of the 2 parabola x -6x-12y51=0. 148. Find the length of the chord of the circle x2+y2+4x+6y-32=0 if its distance from the center of the circle is 5m. Ans. 8.94m 149. Determine the area of the equilateral triangle inscribed in the circle x2+y2-20x+64=0. Ans. units A=46.8 152. What is the equation of the directrix of the curve x2=16 y. Ans. y+4=0 153. The curve has an equation of x2 = Cy +D. the length of the latus rectum is 4 and the vertex is at (0,2). Compute the value of C. Ans. C=4 154. What is the coordinates of the focus of the parabola x2+8x-16+32=0. Ans. focus(-4,6) 155. The axial cross section of the head light reflector is a parabola with the bulb center as focus. Find the depth of the head light in cm, if the bulb’s center is 2cm from the vertex and the radius of the head light is 10 cm. Ans. x= 12.5 cm sq. 150. A parabola passes thru (3,4). The vertex is at the origin and the focus on the x-axis. Find the length of the latus rectum. Ans. 16/3 Downloaded by Michael Berco (bercomichael29@gmail.com) 156. A construction worker built a parabolic hut which is 16.1 m wide at the base and 12.4 m high at the center. How high above the base should a ceiling 12.2 m wide be constructed. Ans. h= 5.28m lOMoARcPSD|13483535 157. The distance between the foci of an ellipse is 5. If its eccentricity is 0.50, what is the distance between the directrices? cm long and lies on the x-axis with its center at the origin. Determine the equation of the ellipse. Ans. 400=0 16x2+25y2- Ans. d= 20 The concrete pavement of the roadway is 3m above the center of the arch. Determine the height above the arch on the roadway 9m from the arch center. Ans. h=4.61m 158. What is the area enclosed by the curve 9x2+25y2225=0. Ans. A=47.124 sq. units 159. Compute the length of the latus rectum of an ellipse x2+2y2+4x+4y+4=0. Ans. L= 1.414 164. Compute the distance between the directrices of the curve 9x2+25y2-54x250y+481=0. Ans. D= 12.5 Ans. Hyperbola 165. Determine the length of the latus rectum of the ellipse having an equation 4x2+9y264x+54y+301=0. Ans. L= 8/3 160. Find the second eccentricity of an ellipse if the distance from the nearest focus to the vertex is 4 and the distance from the farthest focus to the vertex is 16. Ans. e’ = 0.75 161. Find the coordinates of the center of the ellipse 16x2+25y2-128x150y+381=0. Ans. Center( 4,3) 162. What is the eccentricity of the curve 9x2+25y2144x+200y+751=0. 170. A hyperbola has its foci at (-5.0) and (1,0). If one of the vertex of the hyperbola is (-4,0), find the length of the latus rectum. Ans. L=5 166. A satellite orbits around the earth in an elliptical path of eccentricity 0.6 and semi-minor axis of length 12,000 miles. If the center of the earth is at one of the foci, find the maximum altitude of the satellite. 171. The equilateral hyperbola xy=8 has the x-axis and y-axis as asymptote. Determine the eccentricity of the hyperbola. Ans. e= 1.414 Ans. max. a= 24000 miles 167. The distance between the foci of an ellipse is equal to 8 and the second eccentricity is equal to 1.333. Compute the perimeter of the curve. Ans. P=25.91 Ans. e= 0.80 163. The perimeter of an ellipse is 28.448 cm. the major axis is 10 169. The graph of the equation 2x23y2=4 indicates what type of curve? 168. An elliptical bridge was constructed having a height of 12m and a base width of 36m. Downloaded by Michael Berco (bercomichael29@gmail.com) 172. A hyperbola has the equation x2+8x-4y2+64y=256. Determine the center of hyperbola. Ans. (4,8) 173. How far from the axis is the focus F of the hyperbola x 22y2+4x+4y=4=0? Ans. d= 2.73 174. A point moves so that the lOMoARcPSD|13483535 difference between its distances from (0,5) and (0,-5)is 8. Find the equation of the locus of the point. Ans. 9y2-16x2=144 175. What is the vertical asymptote of the curve x3 y= 3 x +3 x 2 +x−3 Ans. x-3=0 176. Find the horizontal asymptote of the curve y= 2x4 x 4 +3 x 2−1 Ans. (-1, 1) 182.) The segment from (-1, 4) to (2,-2) is extended three times its own length. The vertical point is Ans. (11, -20) 183.) The point (a,1), (b,2) and (c,3) are which of the following is true? Ans. c-b=b-a 184.) It he slope other line connecting the origin and point P is 3/4, find the abscissa of P if its ordinate is 6. Ans. x=8 185.) Find the inclination of the line passing through (5-3) and (10, 7) Ans. 14.93 Ans. y-2=0 177. Compute the rectangular coordinates of a point having a polar coordinate of (7,38°). Ans. (5.52,4.31) 178. Find the equation of the plane that contains the point (2,-4,1) and is perpendicular to the vector N equal to 2i3j+4k. Ans. 2x-3y+4z=0 179. A plane has an equation of 4x+y+8z+33=0. Find the distance between the point A (1,5, -3) Ans. d=2 181.) Determine the coordinates of the point which is three-fifths of the way from the post (2-5) to the point (3.5) 186.) Find the angle formed by the lines 2x = y-8=0 and x = 3y = 4 = 0. Ans. 45 187.) Find the angle between the lines 3x + 2y = 6 and x = y = 6. Ans. 11.3099° 188.) What is the acute angle between the lines y = 3x = 2 and y = 4x = 9? Ans. 4.398 189.) Find the distance of line 3x + 4y = 5 from the origin. Ans. (4,-4) 190.) The two points on the lines 2x + 3y + 4= 0 which are at a distance 2 from the line 3x + 4y-6 = 0 are? Ans. 2 units191.) The distance from the point (2.1) to the line 4x - 3y + 5 = 0 is? Ans. (2, 1) 192.) Determine the distance from (5.10) to the line x - y = 0. Ans. 3.54 units 193.) The distance from a point (1.3) to the line 4x + 3y + 12 = 0 is? units Ans. 5 194.) Find the distance between the given lines 4x 3y = 12 and 4x - 3y =-8. Ans. 4 units 195.) Find the distance between the line, 3x + y -12 = 0 and 3x=y-4=0. Ans. 8/√10 196.) What is the length of the line with a slope of 4/3 from a point (6, 4) to the yaxis? Ans. 10 units 197.) Find the slope of the line defined by y-1 x 5. Ans. m=1 198.) What is the slope of the line 3x +2y+1=0? Ans.m= -3/2 199.) In cartesian coordinates the vertices of a triangle are defined by the following points: (-2, 0), (4, 0) and (3.3). What is the area? Ans. 9 square units 200.) Given the three vertices of a triangle whose coordinates are A (1, 1), B (3,-3) and (5,-3). Find the area of the triangle. Ans. 4 square units 201.) In cartesian coordinates, the vertices of a square are (1.1), (0.8), (4.5) and (-3, 4). What is the area? Ans. 25 square units 202.) A line passes thru (1,-3) and -4, 2). What the equation of the line in slope-intercept from. Ans. y= -x-2 203.) What is the intercept of the line passing through (1, 4) and 4, 1)? Ans. x+y=5 204.) Find the equation of a straight line with a slope of 3 and a y-intercept of 1. Ans. 5 Downloaded by Michael Berco (bercomichael29@gmail.com) lOMoARcPSD|13483535 205.) If the point (-2, 3), (x, y) and (-3,-5) lie on a straight line, then the equation of the line is _________________. Ans. 2x+y+1=0 215.) Find the equation of the perpendicular bisector of the line joining (5,0) and (-7,3)? Ans. 8x-2y+11=0 216.) Which of the following line is parallel to the line 3x2y=6 = 0? Ans. 3/2 206.) The equation of a line that intercepts the x- axis at y=-6 is. Ans. 3x-2y=12 207.) A line with an inclination of 450 passes through (-5/29/2). What is the x-coordinate of a point on the line if its coordinates of a point on the line if its corresponding ycoordinate is 6? Ans. x=8 208.) Find the equation of the line passing through the origin and with slope of 6? Ans. y-6x=0 209.) Find the equation of the line if the x = intercept and y=intercept are -2 and 4, respectively. Ans. y-2x-4=0 210.) Determine B such that 3x + 2y - 7 = 0 is perpendicular to 2x - by + 2= 0. Ans.B=3 211.) The line 2x - 3y + 2 = 0 is perpendicular to another line L1 or unknown equation. Find the slope of L1. Ans. -3/2 212.) A line through (-5,2) and (1,-4) is perpendicular to the line through (x,-7) and (8,7), Find x. Ans. -6 213.) What is the equation of the line that passes thru (4.0) and is parallel to the line x-y2 = 0? Ans. y=x-4 214.) Find the equation of the line through point (3, 1) and is perpendicular to the line x = 5y = 5 = 0. Ans. 5x-y=14 217.) The equation of the line through (3-3,-5) parallel to 7x = 2y - 4 = 0 is. Ans. 7x+2y+31=0 218.) What is the equation of the line joining the points (3, -2) and (-7, 6)? Ans. 4x+5y-2=0 219.) What is the equation of the line passing through (2,6) with the x-intercept half the y=intercept? Ans. 2x+y-2=0 220.) Find the slope of a line having a parametric equation of x=2=1 and - 3t. Ans. y=-3y+11 221.) Find the slope of the line having a parametric equation y = 4t = 6 and x= 1. Ans. 4 222.) Two vertices of a triangle are (2.4) and (-2, 3) and the area is 2 square units, the focus of the third vertex is. Ans. 223.) Find the area of the triangle which the line 2x - 3y + 6 = 0 forms with the coordinate axis. Ans. (2, 3) 224.) A line passes through point (2, 2). Find the equation of the line segment intercepted by the coordination axes is the square root of 5. Ans. a=1 ; b=-2 225.) What is the radius of the circle x2=y2 - 6y = 0? Ans. r=3 226.) What are the coordinates of the center of the curve x2=y2 - 2x - 4y - 31 Ans. = 0? (1, 2) 227.) A circle whose equation is x3 = y2 = 4x = 6y - 23 = 0 has its center at. Ans. (-2, -3) 228.) What is the radius of a circle with the equation if: x2 6x = y2 - 4y - 12 = 0. Ans. r=5 229.) The diameter of a circle described by 9x2 + 9y2 = 16 is? Ans. 8/3 230.) How far from the y-axis is the center of the curve 2x2 + 2y2 + 10 x - 6y - 55 = 0? Ans. 2.5 231.) What is the distance between the center of the circles x2 = y2 = 2x = 4y - 3 = 0 and x2 = y2 - 8x - 6y = 7 = 0? Ans.7.071 232.) The shortest distance from A (3, 8) to the circle x2 = y2 = 4x - 6y = 12 is equal to? Ans. x= shortest distance 233.) The equation x2 = y2 4x = 2y - 20 =0 describes: Ans. 5 234.) The center of a circle is at (1, 1) and one point on its circumference is (-1,-3). Find the other end of the diameter, through (-1, -3)? Ans. (3, 5) 235.) Find the area (in square units) of the circle whose equation is x2 = y2 =6x-8y. Ans. 25(3.14) square units 236.) Determine the equation of the circle whose radius is 5, center on the line x = 2 and tangent to the line 3x - 4y = 11 = 0. Ans. 237.) Find the equation of the circle with the center. At (-4,5) and tangent to the line 2x = 7y - 10 = 0. Ans. Downloaded by Michael Berco (bercomichael29@gmail.com) lOMoARcPSD|13483535 249.) Find the eccentricity of 238.) Find the value of k for which the equation x2 = y2 = 4x - 2y - k = 0 represent a point circle. Ans. k=-5 239.) 3x2 = 2x - 5y = 7 = 0. Determine the curve. Ans. parabola 243.) Given the equation of a parabola 3x = 2y2 - 4y = 7 = 0. Locate its vertex. Ans. (-5/3, 1) 244.) In the equation y = x2 = x = 1 where is the curve facing? Ans. facing downward 245.) What is the length of the length of the latus rectum of the curve x2 = 20y? Ans. 4a=20 246.) An ellipse with center at the origin has a length of major axis 20 units. If the distance from center of ellipse to its focus is 5, what is the equation of its directrix? Ans. x=20 247.) What is the length of y +8x-32=0? Ans. 2.7 y - +9 right triangle with adjacent side x, opposite side y and the hypotenuse r. = cosꝊ Ans. 1.80 2 -2 y 2 258.) Find the polar equation of the circle of radius 3 units and center at (3,0). Ans. 2.7 Ans. r=6cosꝊ 251.) The semi-transverse 259.) Given the polar equation r=5sinꝊ. Determine the rectangular coordinates (x,y) of a point in the curve when Ꝋ is 30 degrees. x2 + 9 Ans. (2.17, 1.25) Ans. 3 252.) What is the equation of the asymptote of the 2 hyperbola 2 x 9 + y 4 =1? Ans. 2x-3y=0 253.) The point of intersection of the planes x+5y-2z=9; 3x2y+z=3 and x+y+z=2 is at 254.) What is the radius of the sphere center at the origin that passes the point 8, 1, 6? √ 101 255.) the equation of a sphere with center at (-3, 2, 4) and of radius 6 units is Ans. x 2 + +6x-4y-8z=7 are y=+ 2 x 2 and which passes through (5/2, 3). Ans. Ans. hyperbola 4 x 2 - y 2 -16=0 261.) Find the equation of the hyperbola with vertices at (-4, 2) and (0, 2) and foci at (-5, 2) and (1, 2) Ans. 5 x2 - 4 y 2 +20x+16y-16=0 Ans. (2, 1, -1) Ans. 260.) Find the equation of the hyperbola whose asymptotes 262. The major axis of the elliptical path in which the earth moves around the sun is approximately 186,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth. Answer: 94,550,000 y2 + z2 256.) Find the polar equation of the circle, if its center is at (4,0) and the radius 4 Ans. r-8cosꝊ=0 248.) 4 x 2 - y 2 =16 is the equation of a/an 2 rsin Ꝋ Ans. (1/4, 0) 250.) How far from the x-axis is the focus P of the y 2 =1 is 4 242.) Find the equation of the directrix of the parabola y2 = 16x. Ans. -4 2 -4 2 axis of the hyperbola 241.) Where is the vertex of the parabola x2 = 4(y -2)? Ans. (0, 2) the latus rectum of 4 x x hyperbola x +4x+4y+4=0? 240.) The focus of the parabola y2 = 4x is at? Ans. (4, 0) 2 the curve 9 36x+8y=4 2 257.) What are the x and y coordinates of the focus of the conic section described by the following equation? (Angle Ꝋ corresponds to a Downloaded by Michael Berco (bercomichael29@gmail.com) 263. Find the eccentricity of the curve 9x2 - 4y2 – 36x + 8y = 4. Answer: 1.8 264. Find the equation of the hyperbola whose asymptotes are y = 2x and y = -2x and lOMoARcPSD|13483535 which passé through ( 5 ,3). 2 Answer: 4x2-y2=16 265. Find the equation of the line that passes through the points (-1,0) and (-4,-12). Answer: y+4x=-4 266. What is the equation of the line through the point (3,2) and has xintercept at x = -1? Answer: y=-x-1 267. Find the equation of the line that has an xintercept at x = -4 and y-intercept at y = 5. Answer: 4y-20=5x 268. Given line equation -3x + 5y = 8.Find the x and y intercepts. Answer: ( −8 ,0¿ 3 269. Find the slope intercept form for the line given by its equation x y − =3 4 5 Answer: 5 y= x−15 4 270. For what values of b the point (2,2b) is on the line with equation x – 4y = 6. Answer: −1 2 271. If the point (x,x) is equidistant from (-2,5) and (3,-2), what is the value of x? Answer: 4 272. You wanted to go to your grandparent’s house, which is 300m away to the left from your house. Also, you were thinking of going to your bestfriend’s house which is 200m away to the right from your house. Find the distance from your grandparent’s house to your bestfriend’s house. Answer: 500 273. Your mother told you that you have to go to the grocery store to buy supplies for your home. You are in the building where you work and the nearest grocery store is located 570m to the left from the building you are in. After that, your sister called you, requesting to buy medicine for her. The nearest drug store is 720m to the right of the building. Your friend wanted to come with you and you two agreed that you are just going to meet in the middle of the grocery store and the drug store. What is the midpoint of thr grocery store and the drug store? Answer: 75 274. You are at a five star hotel which is located (2,5) on an XY-coordinate plane and you need to visit a historical monument which is located at (-4,3) for your project. Find the distance between the five star hotel and the historical monument. Answer: 2 √ 10 275. A detective must find the Downloaded by Michael Berco (bercomichael29@gmail.com) coordinates of the vault which is the midpoint between a mall located at (-7,-3) and a wildlife park located at (3,-3). Answer: (-2, -3) 276. The sign (S) which has the coordinates (3,4) on the XY-coordinate plane, divides internally the path of the city school (T), which has the coordinates (-5,1), and the mall (U) with the ratio 2:3. Find the coordinates of the mall. Answer: (15,8.5) 277. The local store (L) with the coordinates (7,-1) and the library (M) with the coordinates (2,3), are two known places in the city. The traffic light (N) for the two areas divide externally the local store and the library in the ratio 1:2. Find the coordinates of the traffic light. Answer: (-8,11) Situation: Given a right triangle with coordinates A(1,5) and C(5,2). 278. What are the coordinates of B? Answer: (1,2) 279. What is the area of the triangle? Answer: 6 sq units Situation: The endpoints of a diameter of a circle are (1,2) and (7,10). 280. What are the coordinates of the center of the circle? Answer: (4,6) 281. Find the area of the circle. Answer: 78.54 sq units lOMoARcPSD|13483535 282. What is the circumference of the circle? Answer: 31.42 units 283. What is the standard equation that describes the graph of the circle? Answer: (x-4)2+(y6)2=25 290. Find the area of tringle ABC. Answer: 12 sq units Situation: Three vertices of a triangle are A(1,2), B(5,10), and C(7,4). Situation: given the coordinates of the circle center (3,2) and B(5,8). 291. What is the equation of the median to segment AC? Answer: y=7x-25 284. What is the equation of the line that touches the circle at point B? Answer: y=3x-7 292. What is the equation of the perpendicular bisector of AC? Answer: y=-3x+15 285. What is the equation of the tangent line that touches the circle at point B? Answer: 293. What is the equation of the altitude to AC? Answer:y=-3x+25 y= −1 29 x+ 3 3 286. What is the area of the region bounded by the xaxis, the y-axis, and the graph 4x – y = 8? Answer: 8 sq units 287. Plot the point P(3,4,5). What is the distance between the origin and point P? Answer: 5 √ 2 288. What is the distance between point P(5,3) and the line 3x + 4y – 7 = 0? Answer: 4 units Situation: Triangle ABC is an equilateral triangle. Point C is (8,0) and A is (0,0). 289. What are the coordinates of point B? Answer: (4 , 4 √ 3) 294. Find the ratio in which the point R(4,24) on the line PQ divides the join of P(2,27) and Q(10,15). Answer: 1:3 295. Find the coordinates of the centroid of the triangle (-4,4), (-2,2), and (6,12). Answer: (0,6) 296. If (3,-1), (2,6), and (-5,7) are the midpoints of the sides of a triangle ABC, find the area of the triangle. Answer: 96 sq units 297. Find the equation of the line passing through (3,4) and having a slope of 2. Answer: 2x-y-2=0 298. Find the equation of the line passing through (4,3) and is Downloaded by Michael Berco (bercomichael29@gmail.com) perpendicular to the line 2x – 5y + 4 = 0. Answer: 5x+2y-14=0 299. Find the equation of a line which makes an angle of 135 degrees with positive direction of x-axis and making an intercept of 3 units on the y-axis. Answer: x+y-3=0 300. Find the length of the sides of the triangle whose vertices are (3,2), (2,1), and (4,-6). Answer: √ 2, √ 53 , √65 301. Find the coordinates of the center of the circle having points (9,3) and (1,-1) as the endpoints of the diameter. Answer: (5,1)