Question

P(−5, 7), A(3, k) and B(k, −1) are given points. If PA = PB, find the value of k.

Answer 1

Given:

Points P(−5, 7), A(3, k), and B(k, −)

Condition: PA = PB

Calculate the distance PA using the distance formula between points

P(−5, 7) and A(3, k):

\[PA = \sqrt{(3 − (−5))^2 + (k − 7)^2}\]

= \[\sqrt{(3+5)^2 + (k − 7)^2}\]

= \[\sqrt{8^2 + (k − 7)^2}\]

= \[\sqrt{64 + (k − 7)^2}\]

Calculate the distance PB using the distance formula between points

P(-5, 7) and B(k, -1):

\[PB = \sqrt{(k - (-5))^2 + (-1 - 7)^2}\]

= \[\sqrt{(k + 5)^2 + (-8)^2}\]

= \[\sqrt{(k + 5)^2 + 64}\]

\[\sqrt{64 + (k - 7)^2}\]   ...[Since PA = PB, we equate the distances.]

= \[\sqrt{(k + 5)^2 + 64}\] 

64 + (k − 7)² = (k + 5)² + 64   ...[Square both sides to eliminate the square root]

\[(k - 7)^2 = (k + 5)^2\]    ...[Subtract 64 from both sides]

(k − 7)² = k² − 14k + 49

(k + 5)² = k² + 10k + 25      ...[Expand both sides]

k² − 14k + 49 = k² + 10k + 25

−14k + 49 = 10k + 25     ...[Subtract k² from both sides]

−14k − 10k = 25 − 49    ... [Rearranging terms]

− 24k = −24  

−14k − 10k = 25 − 49

−24 = k = 1     ...[Divide both sides by]