JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Parabola

If the common tangent to the parabolas

y^{2} = 4 x

and

x^{2} = 4 y

also touches the circle,

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

, then

c

is equal to

(A) $\frac{1}{2 2}$
(B) $\frac{1}{2}$ ✓ Correct
(C) $\frac{1}{2}$
(D) $\frac{1}{4}$

Answer: (B)

Solution

A tangent to

y^{2} = 4 x

can be written

y = m x + \frac{1}{m}

. For it also to touch

x^{2} = 4 y

, substitute to get

x^{2} - 4 m x - \frac{4}{m} = 0

and set the discriminant to zero:

16 m^{2} + \frac{16}{m} = 0

, so

m^{3} = - 1

and

m = - 1

. The common tangent is then

x + y + 1 = 0

, and for it to touch

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

the distance of the centre from the line must equal

c

c = \frac{∣1∣}{1 ^{2} + 1 ^{2}} = \frac{1}{2}

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option

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