JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Circle

Consider the circle

x^{2} + y^{2} = 9

and the parabola

y^{2} = 8 x

. They intersect at

P

and

Q

in the first and the fourth quadrants, respectively. Tangents to the circle at

P

and

Q

intersect the

x

-axis at

R

and tangents to the parabola at

P

and

Q

intersect the

x

-axis at

S

.

The ratio of the areas of the triangles

P QS

and

P QR

Answer: (C)

Solution

Solving

x^{2} + y^{2} = 9

and

y^{2} = 8 x

x^{2} + 8 x = 9

, so

x^{2} + 8 x - 9 = 0

, giving

x = 1

(taking positive root).

Thus

y^{2} = 8

, so

y = \pm 22

.

Points:

P = (1, 22)

Q = (1, - 22)

.

For the circle, tangent at

P

x + 22 y = 9

, intersects x-axis at

R = (9, 0)

.

For the parabola

y^{2} = 8 x

, tangent at

P

y \cdot 22 = 4 (x + 1)

, intersects x-axis at

S = (- 1, 0)

.

Area of

△ P QR = \frac{1}{2} \cdot ∣ P Q ∣ \cdot ∣ distance from R to line PQ ∣ = \frac{1}{2} \cdot 42 \cdot 8 = 162

Area of

△ P QS = \frac{1}{2} \cdot 42 \cdot 2 = 42

Ratio =

42 : 162 = 1 : 4

JEE Advanced · 2007 · 08 Apr 2007 · Paper 1 · Single correct option