The locus of the mirror image of the point (2, 3) in the family of lines (2x - 3y + 4) + k(x - 2y + 3) = 0 for all real values of k is a:

circle of radius sqrt(2)
circle of radius sqrt(3)
straight line parallel to x-axis
straight line parallel to y-axis

Correct answer: circle of radius sqrt(2)

Solution

All lines in the family pass through (1, 2), the intersection of 2x-3y+4=0 and x-2y+3=0. The image of (2,3) in any such line lies on a circle centred at (1,2) with radius = distance((2,3),(1,2)) = sqrt(1²+1²) = sqrt(2).

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