Locus of the image of the point ( 2 , 3 ) in the line 2 x - 3 y + 4 + k x - 2 y + 3 = 0 , k ∈ R , is a

Step-by-step solution

Let image is ( α , β ) ∴ point ( 2 + α 2 , 3 + β 2 ) lies on the given variable line. ∴   2 ( 2 + α 2 ) − 3 ( 3 + β 2 ) + 4 + k ( 2 + α 2 − 2 × 3 + β 2 + 3 ) = 0 ⇒ 2 α − 3 β + 3 + k ( α − 2 β + 2 ) = 0         . . . . . . . 1 again, product of two slopes = − 1 ⇒ 3 − β 2 − α × 2 + k 3 + 2 k = − 1 ⇒ 6 − 2 β + k ( 3 − &#