College Algebra, Chapter 8, 8.1, Section 8.1, Problem 20

Determine the focus, directions and focal diameter of the parabola $\displaystyle x - 7y^2 = 0$. Then, sketch its graph.



The equation $\displaystyle x - 7y^2 = 0; y^2 = \frac{x}{7}$ is a parabola that opens to the right. The parabola has the form $y^2 = 4px$. So


$
\begin{equation}
\begin{aligned}

4p =& \frac{1}{7}
\\
\\
p =& \frac{1}{28}

\end{aligned}
\end{equation}
$


So, the focus is at $\displaystyle (p,0) = \left(\frac{1}{28},0 \right)$ and directrix $\displaystyle x = -p = \frac{-1}{28}$. Also, $\displaystyle 2p = 2 \left( \frac{1}{28} \right) = \frac{1}{14}$, thus the endpoints of the latus rectum are at $\displaystyle \left( \frac{1}{28}, \frac{1}{14} \right)$ and $\displaystyle \left( \frac{1}{28}, \frac{-1}{14} \right)$. The focal diameter is $\displaystyle 4p = 4 \left( \frac{1}{28} \right) = \frac{1}{7} $ units. Therefore, the graph is