Question

select the correct answer. given the focus (-1, 15) and the directrix x = -4, what is the equation of the parabola? a. x = (1/6)(y - 15)² - 5/2 b. y = (1/6)(x - 15)² - 5/2 c. x = - (1/6)(y - 15)² - 5/2 d. x = (1/6)(y + 15)² + 5/2

Explanation:

Step1: Recall parabola definition

For any point $(x,y)$ on parabola, distance to focus equals distance to directrix.

Step2: Set up distance equations

Focus $(-1,15)$, directrix $x=-4$.
Distance to focus: $\sqrt{(x+1)^2+(y-15)^2}$
Distance to directrix: $|x+4|$
Set equal: $\sqrt{(x+1)^2+(y-15)^2}=|x+4|$

Step3: Square both sides

$(x+1)^2+(y-15)^2=(x+4)^2$

Step4: Expand both sides

$x^2+2x+1+(y-15)^2=x^2+8x+16$

Step5: Simplify to solve for $x$

Cancel $x^2$, rearrange terms:
$(y-15)^2 = x^2+8x+16 - x^2-2x-1$
$(y-15)^2=6x+15$
$6x=(y-15)^2-15$
$x=\frac{1}{6}(y-15)^2-\frac{15}{6}$
$x=\frac{1}{6}(y-15)^2-\frac{5}{2}$

Answer:

A. $x = \frac{1}{6}(y - 15)^2 - \frac{5}{2}$