JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Parabola

If the normals of the parabola

y^{2} = 4 x

drawn at the end points of its latus rectum are tangents to the circle

(x - 3)^{2} + (y + 2)^{2} = r^{2}

, then the value of

r^{2}

Answer: 2

Solution

For the parabola

y^{2} = 4 x

, the latus rectum endpoints are

(1, 2)

and

(1, - 2)

.

The slope of the tangent at

(t^{2}, 2 t)

\frac{1}{t}

, so the slope of the normal is

- t

.

At

(1, 2)

where

t = 1

: normal has slope

- 1

, equation

y - 2 = - 1 (x - 1)

, i.e.,

x + y - 3 = 0

.

At

(1, - 2)

where

t = - 1

: normal has slope

1

, equation

y + 2 = 1 (x - 1)

, i.e.,

x - y - 3 = 0

.

The circle has center

(3, - 2)

. The distance from the center to each normal line equals

r

r = \frac{∣3 + ( - 2 ) - 3∣}{1 ^{2} + 1 ^{2}} = \frac{∣ - 2∣}{2} = \frac{2}{2} = 2

Therefore,

r^{2} = 2

JEE Advanced · 2015 · 24 May 2015 · Paper 1 · Integer answer

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