Question

Find the standard form of the equation of the parabola with a focus at (-4, 0) and a directrix at x = 4.

Answer 1

@Michele_Laino @rational Would either of you be able to help me with this question

Answer 2

hint:
given a subsequent parabola:
\[x = a{y^2} + by + c\]
the equation of the directrix, and the coordinate of its focus are:
\[\begin{gathered}
x = - \frac{{1 + {b^2} - 4ac}}{{4a}},\quad \left( {directrix} \right) \hfill \\
\hfill \\
F = \left( {\frac{{1 - {b^2} + 4ac}}{{4a}},\quad - \frac{b}{{2a}}} \right),\quad \left( {focus} \right) \hfill \\
\end{gathered} \]

Answer 3

so we can write this algebraic system:
\[\left\{ \begin{gathered}
- \frac{{1 + {b^2} - 4ac}}{{4a}} = 4 \hfill \\
\hfill \\
\frac{{1 - {b^2} + 4ac}}{{4a}} = - 4 \hfill \\
\hfill \\
- \frac{b}{{2a}} = 0 \hfill \\
\end{gathered} \right.\]

Answer 4

from the third equation we get b=0

Answer 5

now, using b=0, we can rewrite the first two equations as below:
\[\left\{ \begin{gathered}
1 - 4ac = - 16a \hfill \\
1 + 4ac = - 16a \hfill \\
\end{gathered} \right.\]

Answer 6

adding those two equation side by side, we get:
2=-32a
so, what is a?

Answer 7

equations*