Conic sections
1. Given
that the straight line x - 2y = 5 intersects with another straight line y = 2x
– 1 at a point H.
a) Determine
the coordinates of H. (3 marks)
b) A
circle touches the straight line y = 2x – 1 at point H and its center lies on
the straight line x + y + 2 = 0. Find the equation of the circle. (8
marks)
c) Show
that the point F(6 , 2) lies outside the circle described in (b). Hence, find
the length of tangent to the circle in (b) from the point F. (4 marks)
Ans:
a) (-1,-3)
b) (x
- 3)2 – (x + 5)2 = 20
c) 6.16
2. An
ellipse has one vertex at (0, 5) and its foci are (0,3) and (0,-3). Find the
equation of the ellipse and sketch the graph. (3
marks)
Ans:
x2/16 + y2/25 = 1
3. a)
P(x1 , y1), Q(x2 , y2) and R(x,y)
are three points on a circle such that PQ is the diameter. By considering the
gradients of PR and QR,show that the equation of the circle is
(x – x1)(x
– x2)+(y – y1)(y – y2) = 0. (3 marks)
b) The straight line 2x – y – 1 = 0
intersects with another straight line
2x + 3y – 18 = 0 at point K and cuts the
y-axis at point L.
i.
Find the coordinates of K and L
ii.
By using the result in (a), find the
equation of the circle with KL as its diameter. Express your answer in the form
(x - h)2+(y – k)2 = r2 .
iii.
The straight line 2x + 3y -18 = 0 cuts
the x-axis at point F. By using pythagoras’ theorem, determine the length of
tangent to the circle described in (b)(ii) from point F (12 marks)
Ans:
b)i. K(3,4) L(0,-2) ii. (x – 3/2)2 + (y - 1)2
= 45/4 iii. 6.78
4. A
conic has the equation 4x2 + 25y2 – 16x + 50y – 59 = 0.
Determine the standard equation and state the shape of the conic. Sketch the graph
of the conic.
(6
marks)
5. The
centre of a circle lies at the focus of the parabola x2 – 4x – 4y +
16 = 0. If the circle passes through the origin, find the equation of the
circle. (6
marks)
Ans:
(x - 2)2 +
(y - 4)2 = 20
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