JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Circle

A line

y = mx + 1

intersects the circle

(x - 3)^{2} + (y + 2)^{2} = 25

at the points P and Q. If the midpoint of the line segment PQ has x-coordinate

- \frac{3}{5}

, then which one of the following options is correct?

(A) $6 \leq m < 8$
(B) $2 \leq m < 4$ ✓ Correct
(C) $4 \leq m < 6$
(D) $- 3 \leq m < - 1$

Answer: (B)

Solution

The midpoint R of chord PQ has coordinates

(- \frac{3}{5}, \frac{- 3 m}{5} + 1)

.

Since CR is perpendicular to PQ (where C is the center (3, -2)), we have:

m (\frac{- \frac{3 m}{5} + 3}{- \frac{3}{5} - 3}) = - 1

Simplifying:

m^{2} - 5 m + 6 = 0

m = 2 or 3

Both values lie in the range

2 \leq m < 4

JEE Advanced · 2019 · 27 May 2019 · Paper 1 · Single correct option

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