An equation of a parabola is given y2 4y 15x 34 0 Find t

An equation of a parabola is given. y^2 - 4y - 15x + 34 = 0 Find the vertex, focus, and directrix of the parabola.

Solution

Given y^2 - 4y - 15x + 34 = 0

then y^2 - 4y + 4 = 15x - 30

(y - 2)^2 = 15(x - 2)

Shiting origin to (2, 2), the new coordinates (x\', y\') of the point (x, y) are given by
x\' = x - 2 and y\' = y - 2

equation of the parabola with respect to the new origin is
y\'^2 = 15x

Vertex is (0, 0), focus is (15/4, 0) and eqn. of directrix is x\' = -15/4 with respect to the new origin.

with respect to the original system of coordinates,
vertex = (-2, -2),
focus = (-2 -15/4, -2) = (-23/4, -2) and
eqn. of directrix is x - 2 = -15/4 ==> x = -15/4 + 2 = -7/4

x = -7/4