Class 10

Math

All topics

Coordinate Geometry

If the point $P(k−1,2)$ is equidistant from the points $A(3,k)$ and $B(k,5)$, find the values of k.

Point P $(k−1,2)$ is equidistant from the points A$(3,k)$ and B$(k,5)$.

$PA=PB$ or $PA_{2}=PB_{2}$

$⇒(3−k+1)_{2}+(k−2)_{2}=(k−k+1)_{2}+(5−2)_{2}$

$⇒(4−k)_{2}+(k−2)_{2}=1_{2}+3_{2}$

$⇒16−8k+k_{2}+k_{2}−4k+4=1+9$

$⇒2k_{2}−12k+20=10$

$⇒2k_{2}−12k+20−10=0$

$⇒2k_{2}−12k+10=0$

$⇒k_{2}−6k+5=0$

$⇒k_{2}−k−5k+5=0$

$⇒k(k−1)−5(k−1)=0$

$⇒(k−5)(k−1)=0$

$k=1$ or $k=5$.