JEE Main · 2015 · 04 Apr 2015 · Offline · Straight Lines

Locus of the image of the point

(2, 3)

in the line

(2 x - 3 y + 4) + k (x - 2 y + 3) = 0, k \in R

, is a :

(A)straight line parallel to $y$ -axis.
(B)circle of radius $2$ .✓ Correct
(C)circle of radius $3$ .
(D)straight line parallel to $x$ -axis.

Answer: (B)

Solution

Every line in the family

(2 x - 3 y + 4) + k (x - 2 y + 3) = 0

passes through the fixed point where

2 x - 3 y + 4 = 0

and

x - 2 y + 3 = 0

meet, namely

(1, 2)

. Reflecting the fixed point

(2, 3)

across a line that pivots about

(1, 2)

keeps the image a constant distance from

(1, 2)

, equal to the distance between

(2, 3)

and

(1, 2)

. That distance is

1^{2} + 1^{2} = 2

, so the image traces a circle of radius

2

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option

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