JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Circle

The line

2 x - y + 1 = 0

is a tangent to the circle at the point

(2, 5)

and the centre of the circle lies on

x - 2 y = 4

. Then, the radius of the circle is :

(A) $53$
(B) $45$
(C) $54$
(D) $35$ ✓ Correct

Answer: (D)

Solution

The tangent at

(2, 5)

2 x - y + 1 = 0

. The radius at the point of tangency is perpendicular to the tangent, so its slope is

- \frac{1}{2}

. The center

(h, k)

lies on the perpendicular through

(2, 5)

y - 5 = - \frac{1}{2} (x - 2)

, i.e.,

x + 2 y = 12

. The center also lies on

x - 2 y = 4

. Solving:

x + 2 y = 12

and

x - 2 y = 4

gives

2 x = 16

, so

x = 8

and

y = 2

. Center is

(8, 2)

. The radius is the distance from

(8, 2)

(2, 5)

r = (8 - 2)^{2} + (2 - 5)^{2} = 36 + 9 = 45 = 35

JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Single correct option

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