JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Circle

If the circle

x^{2} + y^{2} - 2 g x + 6 y - 19 c = 0

g, c \in R

passes through the point

(6, 1)

and its centre lies on the line

x - 2 cy = 8

, then the length of intercept made by the circle on x-axis is

(A) $11$
(B)4
(C)3
(D) $223$ ✓ Correct

Answer: (D)

Solution

Since the circle

x^{2} + y^{2} - 2 g x + 6 y - 19 c = 0

passes through

(6, 1)

, substitution gives

37 - 12 g + 6 - 19 c = 0

, i.e.

12 g + 19 c = 43

. Its centre is

(g, - 3)

, and requiring this to lie on

x - 2 cy = 8

gives

g + 6 c = 8

. Solving the two relations yields

c = 1

and

g = 2

, so the circle is

x^{2} + y^{2} - 4 x + 6 y - 19 = 0

. Setting

y = 0

to find where it meets the

x

-axis gives

x^{2} - 4 x - 19 = 0

, whose roots are

2 \pm 23

; the intercept they cut off has length

(2 + 23) - (2 - 23) = 223

JEE Main · 2022 · 27 Jul 2022 · Shift 1 · Single correct option

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