Find the equation of the parabola if the vertex is $\left(1,2\right)$ and latus rectum is $y = 5$.
Given: Vertex $= V = \left(h,k\right) = \left(1,2\right)$
Draw a perpendicular from V to the latus rectum.
It passes through the focus F.
$\therefore$ $\;$ F is $\left(1, 5\right)$
Also, $VF = a = 3$
From the given data, the parabola is open upwards.
$\therefore$ $\;$ The equation is of the form
$\left(x - h\right)^2 = 4a \left(y - k\right)$
$\therefore$ $\;$ The required equation is
$\left(x - 1\right)^2 = 4 \times 3 \times \left(y - 2\right)$
i.e. $\left(x - 1\right)^2 = 12 \left(y - 2\right)$