Find the vertex, focus, axis, directrix and lotus - rectum of the following parabolas
y2 = 8x + 8y
Given equation of the parabola is y2 = 8x + 8y
⇒ y2 - 8y = 8x
⇒ y2 - 8y + 16 = 8x + 16
⇒ (y - 4)2 = 8(x + 2)
Comparing with the standard form of parabola (y - a)2 = 4b(x - c) we get,
⇒ 4b = 8
⇒ b = 2
⇒ The vertex is (c, a) = (- 2, 4)
⇒ The focus is (b + c, a) =
⇒ The equation of the axis is y - a = 0 i.e, y - 4 = 0
⇒ The equation of the directrix is x - c = - b
⇒ Directrix is x - (-2) = -2
⇒ Directrix is x = -2 -2
⇒ Directrix is x = -4
⇒ Length of latus rectum is 4b = 8.