Find an equation of a parabola with horizontal axis and vertex at the origin that passes through the point $( - 1,5 )$
A) $y ^ { 2 } = 20 x$
B) $( y - 5 ) ^ { 2 } = x$
C) $x ^ { 2 } = - 25 ( y - 1 )$
D) $( x - 5 ) ^ { 2 } = - 25 y$
E) $y ^ { 2 } = - 25 x$

Question

Quizplus · Accepted Answer

Since the parabola has a horizontal axis and passes through the point (-1,5), we know that the vertex is at the origin and the directrix is the line x = 1. This gives us the equation $x = -\frac{p}{2}y^2$ for some value of $p$. To find $p$, we use the fact that the point (-1,5) is on the parabola: $$-1 = -\frac{p}{2}(5)^2 = -\frac{25}{2}p.$$ Solving for $p$, we get $p = \frac{2}{-25} = -\frac{2}{25}$. Substituting into the equation for the parabola, we get $x = \frac{1}{25}y^2$, which is equivalent to $y^2 = -25x$. Therefore, the correct choice is $\boxed{	ext{(E)}}$.

Find an Equation of a Parabola with Horizontal Axis and Vertex