Question

The length of the latus rectum of a parabola whose axis is parallel to the $y-$ axis and is passing through the points $(0,1),(1,2)$ and $(2,\frac{7}{2}),$ is

Answer 1

Let the equation of the parabola be $y=A{x}^2+Bx+C$ as its axis is parallel to $y$ axis.
It is passing through $(0,1),(1,2)$ and $(2,\frac{7}{2})$
$\therefore C=1$
$\Rightarrow y=A{x}^2+Bx+1$
$A+B=1\dots \left(1\right)$
$8A+4B=5\dots \left(2\right)$
Solving $\left(1\right)$ and $\left(2\right),$ we get
$A=\frac{1}{4},B=\frac{3}{4}$

Now, length of the latus rectum, $L=\frac{1}{|A|}=4$