Question

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

Answer 1

The given points are P(k - 1, 2), A(3, k) and B(k, 5).

∵ AP = BP

∴ AP² = BP²

⇒ (k - 1 - 3)² + (2 - k)² = (k - 1 - k)² + (2 - 5)²

⇒ (k - 4)² + (2 - k)² = (-1)² + (-3)²

⇒ k² - 8k + 16 + 4 + k² - 4k = 1 + 9

⇒ k² - 6k + 5 = 0

⇒ (k - 1) (k - 5) = 0

⇒ k = 1 or k = 5

Hence, k = 1 or k = 5 

Answer 2

It is given that P(k − 1, 2) is equidistant from the points A(3, k) and B(k, 5).
∴ AP = BP

\[\Rightarrow \sqrt{\left[ \left( k - 1 \right) - 3 \right]^2 + \left( 2 - k \right)^2} = \sqrt{\left[ \left( k - 1 \right) - k \right]^2 + \left( 2 - 5 \right)^2} \left( \text{ Distance formula }  \right)\]

\[ \Rightarrow \sqrt{\left( k - 4 \right)^2 + \left( 2 - k \right)^2} = \sqrt{\left( - 1 \right)^2 + \left( - 3 \right)^2}\]

Squaring on both sides, we get 

\[k^2 - 8k + 16 + 4 - 4k + k^2 = 10\]

\[ \Rightarrow 2 k^2 - 12k + 10 = 0\]

\[ \Rightarrow k^2 - 6k + 5 = 0\]

\[ \Rightarrow \left( k - 1 \right)\left( k - 5 \right) = 0\]

\[\Rightarrow k - 1 = 0 \text{ or k }  - 5 = 0\]

k = 1 or k = 5

Thus, the value of k is 1 or 5.