Question

What are the vertex, focus, and directrix of the parabola with the equation x^2 y^2-8x 4y 4=0?

vertex (4, –3); focus (4, –4); directrix y = –2vertex (–4, 3); focus (–4, –1); directrix y = 7vertex (4, –3); focus (4, –7); directrix y = 1vertex (–4, 3); focus (–4, 2); directrix y = 4

Answer 1

The vertex, focus, and directrix of the given parabola are (1, 2), (1, 2), and y = -1, respectively.

Step-by-step explanation:

The equation given is not in the standard form of a parabola, which is y = ax^2 + bx + c. To rewrite the equation in this form, we need to complete the square for both the x and y terms.

First, we group the x terms and the y terms separately and move the constant term to the other side:

Divide the equation by 4 to get:

x^2 - 2x + y^2 - 4y = -1

To complete the square, we add the square of half the coefficient of x and y to both sides:

x^2 - 2x + 1 + y^2 - 4y + 4 = -1 + 1 + 4

Simplify:

(x - 1)^2 + (y - 2)^2 = 4

Now the equation is in the standard form of a parabola, where the vertex is (1, 2), the focus is also (1, 2), and the directrix is the line y = -1.