Mathematics QS 026
Topic 1 : Conic sections – Tutorial
TUTORIAL :CONIC SECTIONS
1. Write the equation of each of the circles that satisfies the stated conditions.
Express the equations in the form x 2  y 2  Dx  Ey  F  0
a. Center at 3,0 and r  3
b. Center at the origin and r  7
c. Tangent to the x - axis , a radius of length 4 and abscissa (x coordinate) of center
is  3 .
d. x intercept of 6, y intercept of  4 and passes through the origin.
e. Tangent to both axes , a radius of 6 and the center in the third quadrant.
2. Find the centre and the length of the radius of each of the circles.
a.
x 2  y 2  6 x  10 y  30  0
b. 4 x 2  4 y 2  4 x  8 y  11  0
3. Find the equations of the tangents to the following circles at the points given.
a) x 2  y 2  2 x  6 y  8  0 , at 2,2
b) x 2  y 2  8 x  2 y  15  0 , at 3,2
c) x 2  y 2  10 y  20  0 , at 2,4
4. Find the equations of the tangents to the circle x 2  y 2  4 x  2 y  24  0
at the points where the circle cuts the line y  x
5. Find the length of the tangents from the point 2,5 to the circle
x 2  y 2  14 x  2 y  34  0
6. Find the length of the tangents from the point 3,1 to the circle
x 2  y 2  4x  8 y  9  0
1
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
7. Find the equation of the circle that passes the origin and has its centre  3,4
8. Find the equation of the circle that passes the three points 1,2 ,  3,8 and  9,6 .
9. Find the coordinates of the points of intersection of each of the following circles with
the corresponding straight lines.
a)
x 2  y 2  8 x  2 y  8  0 and y  7 x  2
b) x 2  y 2  10 x  6 y  31  0 and y  5 x  15
10. Find the coordinates of the points of intersection of each of the following pairs of
circles.
a) x 2  y 2  3x  5 y  4  0 and x 2  y 2  x  4 y  7  0
b) x 2  y 2  5 x  3 y  4  0 and x 2  y 2  4 x  6 y  12  0
c) x 2  y 2  4 x  3 y  5  0 and x 2  y 2  6 x  5 y  9  0
11. Show the circles x 2  y 2  10 x  8 y  18  0 and x 2  y 2  8 x  4 y  14  0 do not
intersect each other.
12. Determine the coordinates of the centre and the radius of the circle with equation
x 2  y 2  2 x  6 y  26  0 .
Find the distance from the point P7,9 to the centre of the circle. Hence find the
length of the tangents from P to the circle.
13. The circle C has equation x 2  y 2  18 x  6 y  45  0
a) Find the radius of C and the coordinates of its centre P .
b) Show that the line y  2 x is a tangent to C and find the coordinates of its point of
contact Q .
c) Show that the tangent to C at the point R6,9 has equation x  2 y  12  0
2
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
d) Given that the tangents to C at Q and R meet at S show that QS  5 .
14. Find the focus, directrix and vertex of each of the following parabolas and sketch
their graphs.
a) x 2  9 y (b) x 2  6 y  0 (c)
y2 = 4x (d) y2 = –8x
15. Find the equation of the parabolas with vertex (0,0) that satisfy the given conditions:
a) directrix x  4  0
b) focus F 0,3
c) Open upward and through passing the point 3,4
16. Sketch the following parabolas showing clearly the focus and directrix of each one.
(a) (y – 2)2 = 4(x – 3)
(b) (y + 2)2 = 8(x – 1)
2
(c) y + 8y = 4x – 12
17. Find the equation of the parabola given,
(a) V (0,2) and F(4,2)
(b)
(c) V(0,0) dan directrix, x – 6=0
(d)
V(0,0) and F (0,-4)
V(0,0) dan directrix, y = -8.
18. Find the equation of each of the following parabola in the standard form.
State the coordinates of their vertices, foci and equations of the directrixs.
Sketch their graphs.
a)
y 2  4 y  6x  8
b)
c)
 x  2 2
 6 y  4
y  2 y  2x  3
2
19. Discuss the parabolas with the equations given below,
(a) y2 – 6y – 8x–7 = 0
(b) y2 – 4y + 8x = 28
(c) x2 + 4x – 4y = 0
(d) y = 3 + 8( x  2)
20.Find the equations of the parabolas that satisfy the following conditions:
a) symmetric to the y-axis and passing through the points 1,1 , 1,3 and  2,0 .
b) symmetric to the y-axis and passing through the points 0,4 , 0,1 and 6,1 .
21. Prove that the line y = 2x + 2 touches the parabola y2 = 16x and find the coordinates
of this point.
22. Prove that the line y = x + 6 cuts the parabola y2 = 32x at two distinct points and find
the coordinates of these two points.
3
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
23. In a suspension bridge the shape of the suspension cable is parabolic. The distance
between the 2 towers are 100 m and the lowest point of the suspension cable is 30 m
below the mid point of the line joining the highest point of the 2 towers. Find the
height of the cable at 15 m and 30 m from the middle of the road.
24. A bridge is to be built in the shape of a parabolic arch. The bridge has a span of 120
m and a maximum height of 25 m. Find the height of the arch at distance 10 m from
the center.
25. A parabolic is formed by revolving a parabola about its axis. A spotlight in the form
of a paraboloid 5 cm deep has its focus 2 cm from the vertex.
Find, to one decimal place, the radius R of the opening of the spotlight.
26. Find the vertices, foci and center for the ellipses.
a)
x2 y 2

1
25 4
4x2 4 y 2

1
b)
9
25
c)
2x² + 3y² = 5
d) x² + 4y² – 6x + 8y = 1
27. P( 2,1 ) is a point on an ellipse with foci ( 3 , 0 ) and (– 3 ,0 ).Find the equation of
this ellipse.
28. Find the equation of an ellipse with one foci is F ( 0,3 ), with equation of the minor
axis x = 2 and the length of the major axis is 12 units.
29. Given the foci of an ellipse are ( –1,6 ) and (–1,0 ),with the length of the major axis is
10 units, find the equation of the ellipse, sketch the ellipse.
x2 y2
30. If the line y  mx  c touches the ellipse 2  2  1 , prove that a 2 m 2  b 2  c 2
a
b
31. By completing the square, show that each of the following equation represents a
hyperbola.
a. 4x2 – y2 – 8x – 6y – 9 = 0
b. 2x2 – y2 – 8x – 4y – 3 = 0
c. 3x2 – y2 – 12x – 4y – 10 = 0
4
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
32. Find the equation of the following hyperbola with
a) Hyperbola with foci (0, 0) and (0, 4) that passes through (12, 9)
b) Hyperbola with a vertex at (0, ±4) and a focus at (0, -5)
c) Hyperbola with asymptotes 2x ± 4y = 0 and the vertex at (8, 0)
33. Sketch the graph of the given equation y 2  4 y  4 x 2  8 x  4
34. Find in Cartesian form, of each of the following parametric equations.
a) x  2t 2 , y  4t
3
b) x  t , y  3t
2
35. For each of the following curves, obtain its parametric equations
a) y 2  6 x
b) 3 y 2  8( x  1)
5
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
ANSWER
1 a) x2 + y2 – 6x = 0
b) x2 + y2 – 49 = 0
c) x2 + y2 + 6x – 8y + 9 = 0 , x2 + y2 + 6x + 8y + 9 = 0
d) x2 + y2 – 6x + 4y = 0
e) x2 + y2 + 12x + 12y + 36 = 0
2 a) C (3, 5); r = 2
1
b) C ( , 1); r  2
2
3 a) x = y
b) x + y = 1
c) y = -2x
4) 2x + 5y + 28 = 0 , 5x + 2y – 21 = 0
5) 5
6)
23
7) x2 + y2 + 6x + 8y = 0
8) 11x2 + 11y2 + 100x – 58y – 39 = 0
9) a) (-1, -5). (0, 2)
b) (-1, 10), (-4, -5)
10) a) (2, 1), (-1, -5)
b) (5, 1), (2, 2)
3 1
c) ( ,  ), (1, -1)
2 2
12) C (-1, 3), r = 6 ; 10 , 8
13) a) 3 5 , P(9, 3)
b) Q (3, 6)
9
9
), Directrix, y = 
4
4
,V(0, 0)
3
3
b) F (0,  ), Directrix, y =
2
2
,V (0, 0)
c) F (1, 0) , Directrix, x = -1
,V (0, 0)
14) a) F (0,
6
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
d) F (-2, 0), Directrix, x = 2
,V (0, 0)
15) a) y2 = 16x
b) x2 = -12y
c) x2 =
9
y
4
16) a) F (4, 2) ,
x=2
b) F (3, -2), x = -1
c) F (0, -4), x = -2
17 a) (y – 2)2 = 16x
b) x2 = -16y
c) y2 = 24x
d) x2 = -32y
18 a) (y – 2)2 = 6(x + 2) , V (-2, 2), F ( 
1
, 2) , Directrix,
2
b) ( x  2) 2  6( y  4) , V (2 , 4) , F (2 ,
11
)
2
Directrix,
3
c) ( y  1) 2  2( x  2) , V (2 ,1) , F ( ,1) , Directrix,
2
x  4
19 a) V (2 , 3) , F (0 , 3) , Directrix,
b) V ( 4 , 2) , F (2 , 2) , Directrix,
x6
c) V (2 ,  1) , F (2 , 0) , Directrix,
d) V (2 , 3) , F (0 , 3) , Directrix,
y  2
x  4
20 a) ( y  1) 2  x  3
3
25
b) ( y  ) 2  ( x  )
2
4
21) (1, 4)
22) (2,8) , (18, 24)
23) 2.7 m , 10.8 m
7
x
x
7
2
y
5
2
5
2
Mathematics QS 026
Topic 1 : Conic sections – Tutorial
24) 24.31 m
25) R  40  6.3 cm
F1 ( 21 , 0) , F2 ( 21 , 0) , C (0,0)
26 a) V1 (5 , 0) , V2 (5 , 0) ,
5
5
b) V1 (0 , ) , V2 (0 ,  ) ,
F1 (0 , 2) , F2 (0 ,  2) , C (0,0)
2
2
5
5
5
5
c) V1 (
, 0) , V2 (
, 0) ,
F1 ( , 0) , F2 (
, 0) , C (0,0)
2
2
6
6
21
21
d) V1 (3  14 ,  1) , V2 (3  14 ,  1) ,
F1 (3 
,  1) , F2 (3 
,  1) , C (3,1)
2
2
x2 y2
27)

1
6
3
( x  2) 2 ( y  3) 2
28)

1
36
32
( x  1) 2 ( y  3) 2
29)

1
16
25
( x  1) 2 ( y  3) 2
31 a)

1
1
4
( x  2) 2 ( y  2) 2
b)

1
7
7
2
( x  2) 2 ( y  2) 2

1
c)
6
18
( y  2) 2 x 2
32 a)

1
1
3
y2 x2
b)

1
16 9
x2 y2
c)

1
64 16
34 a) y 2  8 x
b) y 2  6 x
3
35 a) x  t 2 , y  3t
2
2
4
b) x  t 2  1 , y  t
3
3
8