What is the standard form of the equation of the parabola with a directrix at x=-8 and a focus at (-7,3)?

Question

Abigail Allen · Accepted Answer

#(y-3)^2=-4(15/2)(x-1/2)#

The directrix is x = 8 the focus S is (-7, 3), in the negative direction of x-axis, from the directrix..

Using the definition of the parabola as the locus of the point that is equdistant from the directrix and the focus, its equation is

#sqrt((x+7)^2+(y-3)^2)=8-x, > 0#,

as the parabola is on the focus-side of the directrix, in the negative x-direction.

Squaring, expanding and simplifying, the standard form is.

#(y-3)^2=-4(15/2)(x-1/2)#.

The axis of the parabola is y = 3, in the negative x-direction and the vertex V is (1/2, 3). The parameter for size, a = 15/2.,

Eva Abramson · Answer

undefined The standard form of the equation of a parabola with a horizontal axis of symmetry, directrix at x = p, and focus at (h, k) is:

(x - h)^2 = 4p(y - k)

Given that the directrix is x = -8 and the focus is (-7, 3), we can identify that h = -7, k = 3, and p = 1 (since the focus is one unit to the right of the directrix).

Substituting these values into the standard form equation:

(x + 7)^2 = 4(1)(y - 3)

Expanding and simplifying:

(x + 7)^2 = 4y - 12

Therefore, the standard form of the equation of the parabola is:

(x + 7)^2 = 4y - 12