The focus of a parabola is (-10, -7), and its directrix is x = 16. Fill in the missing terms and signs in the parabola's equation in standard form. (y )^2= (x )

ANSWER

[tex](y + 7)^2= - 52(x-3)[/tex]

It was given that the parabola has its focus at:

(-10,-7)

and directrix at:

x=16.

We need to determine the vertex of this parabola which is midway between the focus and the directrix.

Therefore the vertex will be at,

[tex]( \frac{16 + - 10}{2} , - 7)[/tex]

[tex](3, - 7)[/tex]

The equation of this parabola is of the form:

[tex](y-k)^2=4p(x-h)[/tex]

where p is the distance from the vertex to the focus.

[tex] |p| = 16 - 3 = 13[/tex]

Since the parabola opens towards the negative direction of the x-axis,

[tex]p = - 13[/tex]

We substitute the vertex and the value for p to get;

[tex](y - - 7)^2=4( - 13)(x-3)[/tex]

[tex](y + 7)^2= - 52(x-3)[/tex]