Question

Find an equation of the ellipse. center: (0, 0) focus: (5, 0) vertex: (6, 0)

Answer 1

Given,
centre : (0,0)
focus : (5, 0)
vertex :(6, 0)

Let S(5, 0) and S'(-5,0) and P (x,y)
SP + S'P = 2a
√{(x-5)² + y²} + √{x+5)² + y²} = 12
√{(x-5)² + y²} = 12 - √{(x+5)² + y²}
fake square both sides,
(x -5)² + y² = 144 + (x+5)² + y² -24 √{(x+5)² + y²}
(x -5)² - (x +5)² = 144 -24 √{(x+5)² + y²}
(-20x -144)/24 = √{(x+5)² + y²}
(-5/6x - 6)² = (x + 5)² + y²
(5x + 36)²/(6)² = (x +5)² + y²
y² + {(6x +30)² - (5x +36)²}/(6)² = 0
y² + {11(x²- 6²)}/6² = 0
y² + x²/(6/√11)² = 11
y²/(√11)² + x²/6² = 1