Conics

QS025/DM035


CONICS

PAST YEARS’ QUESTIONS

Mid Semester Exam

1. Express the equation of the ellipse 4x2 + y2 – 8x + 4y + 4 = 0 in the standard form. Hence,
determine the coordinates of the centre and the vertices on the major axis. (2006/2007)

2. A Circle with centre (5, 8) touches the y-axis and passes through the point (2, y). Find the
general equations of the circle. Determine the possible values of y. (2006/2007)

3. Given the circle C1 : x2 + y2 – 10x + 18y + 70 = 0 and C2 : x2 + y2 – 6y – 7 = 0.
Find the centers and radius of both circles, and hence show that these two circles do not
touch each other. Determine the shortest distance between them. (2005/2006)

4. A circle with center (1, 5) touches the x-axis. Find the equation of this circle in a general
form. (2004/2005)

5. Find the equation of the parabola with focus (1, 2) and the directrix is the x- axis. Sketch the
graph of the parabola. (2004/2005)

6. Find the coordinate of vertices and foci of the following of conic section given
3(x + 4)2 + 4(y – 2)2 = 12 (2003/2004)

7. Find the coordinate of focus and directrix equation of parabola x2 = 6y. Hence sketch the
graph. (2002/2003)

8. Find the coordinate of vertices and foci of the following of conic section given 2x2 + 8y2 = 32
(2001/2002)

9. Express the equation of parabola y2 + 4y – 12x – 8 = 0 in the standard form. Hence,
determine the vertex and the focus of the parabola. (2007/2008)

10. A circle touches the line 5x + y = 3 at the point (2, 7) and its center lies on the line
x – 2y = 19. Find the point of intersection between the normal to the circle at (2, 7) and the
line x – 2y = 19. Hence, determine the center and the standard equation of the circle.
(2007/2008)

11. A circle passes through the point (5, 2) and touches the line y + x = 9 at the point (3, 6). Find
the coordinates of the center and the radius of the circle. Hence, state the standard equation of
the circle. (2008/2009)




1

, QS025/DM035


12. The equation of the circle P is given by x2 + y2 – 4x + 6y – 12 = 0.
(a) Find the coordinates of its center and radius.
(b) Find the perpendicular distance from the centre of P to the line 3x + 4y = k in terms of k,
where k is a constant.
(c) Hence, find the values of k such that the line 3x + 4y = k is a tangent to the circle.
(2009/2010)

13. The major vertices of an ellipse are (1, 2) and (9, 2). The distance between the two foci of
the ellipse is 8 units. Find the standard equation of the ellipse. (2010/2011)


ANSWERS

1. Centre : (1, 2) Vertices : (1, 0) and (1, 4)
2. x2 + y2 – 10x – 16y + 64 = 0 ; y = 4 or y = 12
3. C1(5, -9) r1 = 6 C2(0, 3) r2 = 4 d > r1r2 = 13 > 0
Shortest distance = 3 units
4. x2 + y2 + 2x – 10y + 1 = 0
y
5. (x + 1)2 = 4(y – 1)
F 2




  
1
6. C(4, 2) V : (6, 2), (2, 2) F: (3, 2), (5, 2)
 4,2  3 ,  4, 2  3
 3
7. F:  0,  Directrix: y  

  
3
 2 2
8. V: (4, 0), (4, 0) F : 2 3, 0 ,  2 3 ,0
(0, 2), (0, 2)
9. (y + 2)2 = 12(x + 1), V(1, 2), F(2, 2)
10. (7, 6), C(7, 6), (x – 7)2 + (y + 6)2 = 26
11. (2, 1), r = 7.071, (x + 2)2 + (y – 1)2 = 50
6k
12. a) C(2, 3), r = 5, b) c) k = 31, k = 19

x  4 2   y  2 2  1
5
13.
25 9




2