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Question

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


Solution

Point P $$(k-1, 2)$$ is equidistant from the points A$$(3, k)$$ and B$$(k, 5)$$.
$$PA=PB$$ or $$PA^2=PB^2$$
$$\Rightarrow (3-k+1)^2+(k-2)^2=(k-k+1)^2+(5-2)^2$$
$$\Rightarrow (4-k)^2+(k-2)^2=1^2+3^2$$
$$\Rightarrow 16-8k+k^2+k^2-4k+4=1+9$$
$$\Rightarrow  2k^2-12k+20=10$$
$$\Rightarrow  2k^2-12k+20-10=0$$
$$\Rightarrow  2k^2-12k+10=0$$
$$\Rightarrow  k^2-6k+5=0$$
$$\Rightarrow  k^2-k-5k+5=0$$
$$\Rightarrow  k(k-1)-5(k-1)=0$$
$$\Rightarrow  (k-5)(k-1)=0$$
$$k=1$$ or $$k=5$$.

Mathematics
RS Agarwal
Standard X

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