Question

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

• A. Straight line parallel to y-axis
• B. Circle of radius $\sqrt{2}$
• C. Circle of radius $\sqrt{3}$
• D. Straight line parallel to x-axis

Answer 1

The correct option is B

Circle of radius $\sqrt{2}$

$(2x-3y+4)+k(x-2y+3)=0$ is a family of equation passing through $T(1,2)$

Let the point be $P(2,3)$ and its image be $P^{\prime }(h,k)$

$PT=P^{\prime }T$

$\Rightarrow P{T}^2=(P^{\prime }T{)}^2$

$\Rightarrow (h-1{)}^2+(k-2{)}^2=(2-1{)}^2+(3-1{)}^2$

$\Rightarrow (h-1{)}^2+(k-2{)}^2=1+1=2$

$\Rightarrow (x-1{)}^2+(y-2{)}^2=2$

Thus, locus is a circle whose radius is $\sqrt{2}$