Question

Write the equation for a parabola with a focus at (−4,8)(-4,8)(−4,8)left parenthesis, minus, 4, comma, 8, right parenthesis and a directrix at x=−6x=-6x=−6x, equals, minus, 6.

Answer 1

Given:

The focus of the parabola is at (-4,8).

The directrix is at x=-6.

To find:

The equation of the parabola.

Solution:

The directrix is at x=-6, which is a vertical line. So, the parabola is horizontal.

The equation of a horizontal parabola is

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

Where, (h,k) is vertex, (h+p,k) is focus and x=h-p.

The focus of the parabola is at (-4,8).

$(h+p,k)=(-4,8)$

$h+p=-4$             ...(i)

$k=8$

The directrix is at x=-6.

$h-p=-6$            ...(ii)

Adding (i) and (ii), we get

$2h=-10$

$h=-5$

Putting h=-5 in (i), we get

$-5+p=-4$

$p=-4+5$

$p=1$

Putting h=-5, k=8 and p=1 in the standard form of the parabola.

$(y-8)^2=4(1)(x-(-5))$

$(y-8)^2=4(x+5)$

Therefore, the required equation of the parabola is $(y-8)^2=4(x+5)$.

Answer 2

Answer:

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Step-by-step explanation: