How do you write the equation of the directrix of the conic section shown below x^2-8x+8y+8=0

Answer 1

Given equation x ² - 8x + 8y + 8 = 0

We first put the equation in to the form for a translated parabola (x - h )² = 4p (y - k)

To do this we complete the squre on the x  terms and move the other terms to right.

x ² - 8x = - 8y - 8

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

x coefficient is 8,(half the x coefficient)² = 16

x ² - 8x + 16 = - 8y - 8 +16

(x - 4)² = - 8y + 8

(x - 4)² = - 8 (y - 1)

(x - 4)² = 4(-2)(y - 1)

Compare it to  parabola equation is (x - h)² = 4p (y - k), where (h, k) = vertex and p = directed distance from vertex to focus.

p = - 2

Vertex of parabola  = (4, 1)

Equation of directrix is y  = k - p

y  = 1 - (-2)

y  = 3

Directrix  y = 3