Answer
Equation of the conic section:
$x = - \frac{1}{{44}}{\left( {y + 1} \right)^2} + 4$
Work Step by Step
From Exercise 49, we see that a parabola with vertex at the origin and directrix $x=-c$ has equation $x = \frac{1}{{4c}}{y^2}$.
If the vertex is translated to $\left( {4, - 1} \right)$, the equation becomes $x - 4 = \frac{1}{{4c}}{\left( {y + 1} \right)^2}$ and the directrix $x=-c$ is shifted $4$ units to the right to be $x=15$. So, $c=-11$.
Substituting $c=-11$ in the equation, we get $x - 4 = - \frac{1}{{44}}{\left( {y + 1} \right)^2}$.
$x = - \frac{1}{{44}}{\left( {y + 1} \right)^2} + 4$
