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Locus of the image of the point \[\left( 2,3 \right)\]in the line\[\left( 2x-3y+4 \right)+k\left( x-2y+3 \right)=0,\]\[k\in R,\] is a : [JEE Main Solved Paper-2015 ]

A) circle of radius \[\sqrt{2}\]

B) circle of radius \[\sqrt{3}\]

C) straight line parallel to x - axis

D) straight line parallel to y - axis

Correct Answer: A

Solution :

The given variable lines is the family lines intersecting at (1, 2) The mirror image P of the given point P and the point P is equidistant from the point (1, 2), hence locus of P is a circle with centre (1, 2) and radius \[PQ=\sqrt{2}\]