Locus of the image of the point (2, 3) in the line `(2x-3y""+""4)""+""k""(x-2y""+""3)""=""0,""kepsiR` , is a :
(1) straight line parallel to x-axis. (2) straight line parallel to y-axis (3) circle of radius `sqrt(2)` (4) circle of radius `sqrt(3)`

Options

A. circle of radius `sqrt(2)`
B. circle of radis `sqrt(3)`
C. straight line parallel to x-axis
D. straight line parallel to y-axis

Answer

A

Explanation

Let Q `(alpha, beta)` be the image of the point (2,3) in the line.
`(2x -3y+4) +k(x -2y + 3)=0`

The equation `(2x-3y + 4) + k (x-2y + 3)=0` represents a family of lines passing through the intersection of `2x - 3y + 4=0` and `x -2y + 3=0` i.e, the poitn A(1,2) . The image of P(2,3) in the line (i) is the point Q which varies in such a way that `AQ= AP`
i.e., `(alpha - 1)^(2) + (beta -2)^(2) = 2`
Hence, the locus of Q is `(x-1)^(2) + (y-2)^(2)` = `(sqrt(2))^(2)`, which is cirle of radius `sqrt(2)`