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300 Problems In Analytic Geometry Section 1
Engineering Managment (Western Institute of Technology)
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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
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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
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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
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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
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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
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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)
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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
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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
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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)