pls slove the math step by step.

Question

Describe the graph of the curve y² - 8x - 6y - 23 = 0 and find the equation axis, tangent at vertex and equation of its directrix.

Answer 1

As the y term is squared and the x term is linear, this is a parabola.

y^2 - 6y = 8x + 23

Complete the square

y^2 - 6y + 9 = 8x + 23 + 9

(y-3)^2 = 8x + 32 = 8(x+4)

To determine the directrix and focus, we write

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

The vertex is (-4, 3)

The directrix is x = -4 -2

The directrix is x = -6

The focus is (-4 + 2, 3) = (-2, 3)

The axis of symmetry is y = 3