JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Circle

A triangle is formed by the tangents at the point

(2, 2)

on the curves

y^{2} = 2 x

and

x^{2} + y^{2} = 4 x

, and the line

x + y + 2 = 0

. If r is the radius of its circumcircle, then

r^{2}

is equal to

Answer: 10

Solution

P (2, 2)

the tangent to

y^{2} = 2 x

2 y = x + 2

, i.e.

x - 2 y + 2 = 0

, and the tangent to

x^{2} + y^{2} = 4 x

y = 2

. With the third side

x + y + 2 = 0

, the vertices are

P = (2, 2)

Q = (- 4, 2)

and

R = (- 2, 0)

. The side lengths are

P Q = 6

QR = 22

and

R P = 25

, and the area of the triangle is

6

. The circumradius satisfies

r = \frac{P Q \cdot QR \cdot R P}{4 \times area} = \frac{6 \cdot 2 2 \cdot 2 5}{24} = 10

, so

r^{2} = 10

JEE Main · 2023 · 29 Jan 2023 · Shift 2 · Integer answer

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