Find the vertex, focus, axis, directrix and lotus - rectum of the following parabolas

Question

Find the vertex, focus, axis, directrix and lotus - rectum of the following parabolas y 2 = 5x - 4y – 9

Philoid · Accepted Answer

Given equation of the parabola is y² = 5x - 4y - 9

⇒ y² + 4y = 5x - 9

⇒ y² + 4y + 4 = 5x - 5

⇒ (y + 2)² = 5(x - 1)

Comparing with the standard form of parabola (y - a)² = 4b(x - c) we get,

⇒ 4b = 5

⇒

⇒ The vertex is (c, a) = (1, - 2)

⇒ The focus is (b + c, a) =

⇒ The equation of the axis is y - a = 0 i.e, y + 2 = 0

⇒ The equation of the directrix is x - c = - b

⇒ Directrix is

⇒ Length of latus rectum is 4b = 5.