Locus of the image of the point (2, 3) in the line 

$(2x-3y+4)+k(x-2y+3)=0,k\in R,\text{ is a }$

The line $(2x-3y+4)+k(x-2y+3)=0,k\in R$

passes through the intersection of line

and $\left.\begin{array}{c}2x-3y+4=0\\ x-2y+3=0\end{array}\right\}$   (A)

for different values of $k$.

Lines given by (A) meet at (1, 2)

Let image of$A(2, 3)$in the family of lines be $B(\alpha ,\beta )$.

Thus (1, 2) is a fixed point for given family of lines, we 

have $AP=BP\Rightarrow (\alpha -1{)}^2+(\beta -1{)}^2=2$

The locus generalises to $(x-1{)}^2+(y-1{)}^2=2$

Thus it is a circle of radius $\sqrt{2}$.