Question

If the point $P(k-1,2)$ is equidistant from the points $A(3,k)$ and $B(k,5)$, then how many values of $k$ are obtained?

Write $0$, if the value cannot be determined.

Answer 1

Point $P(k-1,2)$ is equidistant from $A(3,k)$ and $B(k,5)$
Distance between $P$ and $A$:
Distance $=$ $D_A$ = $\sqrt{(k-1-3{)}^2+(2-k{)}^2}$
Distance between P and B:
Distance = $D_B$ = $\sqrt{(k-1-k{)}^2+(2-5{)}^2}$
Now, $D_A=D_B$
$\Rightarrow \sqrt{(k-1-3{)}^2+(2-k{)}^2}$ = $\sqrt{(k-1-k{)}^2+(2-5{)}^2}$
$\Rightarrow k^2+16-8k+4+k^2-4k=1+9$
$\Rightarrow 2k^2-12k+10=0$
$\Rightarrow k^2-6k+5=0$
$\Rightarrow (k-5)(k-1)=0$
$\Rightarrow k=1,5$