Question

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

Answer 1

Given that, the vertex is $(0, 0)$ and the parabola is symmetric about the $y-axis$,

the equation of the parabola is either of the form $x^2= 4ay$ or $x^2= -4ay.$
The parabola passes through the point $(5, 2)$, which lies in the first quadrant.
Therefore, the equation of the parabola is of the form $x^2= 4ay$, while point $(5, 2)$ must satisfy the equation $x^2= 4ay$

$∴ 5^2 = 4a(2)$

$25 = 8a$

$a = \dfrac{25}{8}$
Thus, the equation of the parabola is

$x^2 = 4 (\dfrac{25}{8})y$

$x^2 = \dfrac{25y}{2}$

$2x^2 = 25y$

∴ The equation of the parabola is $2x^2 = 25y$ (Ans.)