Question

Find the equation of the parabola that satisfies the following conditions: Vertex $(0,0)$ passing through $(5,2)$ and symmetric with respect to $y$- axis

Answer 1

Since the vertex is $(0,0)$ and the parabola is symmetric about the $y$-axis. Equation of the parabola is either of the form $x^2=4ay$ or $x^2=-4ay$
The parabola passes through point $(5,2)$ which lies in the first quadrant
Therefore, the equation of the parabola is of the form $x^2=4ay$
Point $(5,2)$ must satisfy the equation $x^2=4ay$
$\therefore {\left(5\right)}^2=4\times a\times 2$
$\Rightarrow 25=8a$
$\Rightarrow a=\frac{25}{8}$
Thus, the equation of the parabola is
$x^2=4\left(\frac{25}{8}\right)y$
$2x^2=25y$