Locus of the image of the point (2,3) in the line (2x-y+4)+k(x-2y+3)=0, k∈R, is a straight line parallel to y-axis, circle of radius √2, straight line parallel to x-axis, or circle of radius √3?

Given family of line is,(2x-y+4)+k(x-2y+3)=0
2x-y+4=0 (i) and x-2y+3=0 (ii)
After solving these two equations we get x=1 and y=2
Now the given point is (2,3)
Let the image of this point in the given line is P(h,k)
So the distance of point P and Q from (1,2) will be same,
Distance Formula = √(x₂-x₁)²+(y₂-y₁)²
⇒√(2-1)²+(3-2)² = √(h-1)²+(k-2)²
⇒(h-1)²+(k-2)²=2
Hence, the locus of P(h,k) is circle with radius √2