Answer:
k = 3
Step-by-step explanation:
Using the distance formula
d = √ (x₂ - x₁ )² + (y₂ - y₁ )²
with (x₁, y₁ ) = A (- 2, 4 ) and (x₂, y₂ ) = P (2k, k)
AP = [tex]\sqrt{(2k+2)^2+(k-4)^2}[/tex]
Repeat
with (x₁, y₁ ) = B (7, - 5) and P = (2k, k)
BP = [tex]\sqrt{(2k-7)^2+(k+5)^2}[/tex]
Given that AP = BP, then
[tex]\sqrt{(2k+2)^2+(k-4)^2}[/tex] = [tex]\sqrt{(2k-7)^2+(k+5)^2}[/tex]
Square both sides
(2k + 2)² + (k - 4)² = (2k - 7)² + (k + 5)² ← expand factors on both sides
4k² + 8k + 4 + k² - 8k + 16 = 4k² - 28k + 49 + k² + 10k + 25
Simplify both sides by collecting like terms
5k² + 20 = 5k² - 18k + 74 ( subtract 5k² from both sides )
20 = - 18k + 74 ( subtract 74 from both sides )
- 54 = - 18k ( divide both sides by - 18 )
k = 3
