Answer:
(- 7, 0 )
Step-by-step explanation:
let the point on the x- axis be (x, 0 )
Then the distance from (x, 0) to (2, - 5) and (- 2, 9) is equal
Use the distance formula to calculate the distances
d = √ (x₂ - x₁ )² + (y₂ - y₁ )²
with (x₁, y₁ ) = (2, - 5) and (x₂, y₂ ) = (x, 0)
d = [tex]\sqrt{(x-2)^2+(0+5)^2}[/tex]
= [tex]\sqrt{(x-2)^2+5^2}[/tex] = [tex]\sqrt{(x-2)^2+25}[/tex]
Repeat with
(x₁, y₁ ) = (- 2, 9) and (x₂, y₂ ) = (x, 0)
d = [tex]\sqrt{(x+2)^2+(0-9)^2\\}[/tex]
= [tex]\sqrt{(x+2)^2+(-9)^2}[/tex] = [tex]\sqrt{(x+2)^2+81}[/tex]
Equating both distances
[tex]\sqrt{(x-2)^2+25}[/tex] = [tex]\sqrt{(x+2)^2+81}[/tex]
Squaring both sides
(x - 2)² + 25 = (x + 2)² + 81 ← expand parenthesis on both sides
x² - 4x + 4 + 25 = x² + 4x + 4 + 81, that is
x² - 4x + 29 = x² + 4x + 85 ← subtract x² + 4x from both sides
- 8x + 29 = 85 ( subtract 29 from both sides )
- 8x = 56 ( divide both sides by - 8 )
x = - 7
Thus the point on the x- axis is (- 7, 0 )
