Find the equation of the parabola if the vertex is $\left(3, -1\right)$, open rightward and the distance between latus rectum and the directrix is $4$.
The required parabola is open rightward.
$\therefore$ $\;$ Let its equation be
$\left(y - k\right)^2 = 4 a \left(x - h\right)$
Vertex is $= V \left(h, k\right) = \left(3, -1\right)$
Distance between latus rectum and directrix $= 2a$
Given: $\;\;$ $2a = 4 \implies a = 2$
$\therefore$ $\;$ Equation of parabola is
$\left(y + 1\right)^2 = 4 \times 2 \times \left(x - 3\right)$
i.e. $\left(y + 1\right)^2 = 8 \left(x - 3\right)$