Question:
Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for $y^{2}=10 x$
Solution:
The given equation is $y^{2}=10 x$.
Here, the coefficient of $x$ is positive. Hence, the parabola opens towards the right.
On comparing this equation with $y^{2}=4 a x$, we obtain
$4 a=10 \Rightarrow a=\frac{5}{2}$
$\therefore$ Coordinates of the focus $=(a, 0)=\left(\frac{5}{2}, 0\right)$
Since the given equation involves $y^{2}$, the axis of the parabola is the $x$-axis.
Equation of directrix, $x=-a$, i.e., $x=-\frac{5}{2}$
Length of latus rectum = 4a = 10