x^2 − 6x + 12y + 57 = 0

Question

find the vertex, focus, and directrix.

Answer 1

Given equation x ^2 - 6x + 12y + 57 = 0

x ^2 - 6x = -12y - 57

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x coefficient is -6,(half the x coefficient)² = 9

So add 9 to each side.

x ^2 - 6x + 9 = -12y - 57 + 9

(x - 3)^2 = -12y - 48

(x - 3)^2 = -12(y + 4)

(x - 3)^2 = 4(-3)(y - (-4))

Compare it to parabola equation (x - h )^2 = 4a( y - k )

a < 0

a = -3

Vertex of parabola (h , k ) = (3, -4)

Focus = (h , k + a ) = [3, -4+(-3)] = (3, -7)

Equation of directrix is y  = k - a

y  = -4 -(-3)

y  = -4+3

y  = -1

Vertex (3, -4), focus (3, -7) and directrix  y = -1.