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

Since the parabola is symmetric about y-axis and has its vertex at the origin, the equation is of the form x² = 4ay or x² = -4ay, where the sign depends on whether the parabola opens upwards or downwards. But the parabola passes through (5, 2) which lies in the first quadrant, it must open upwards. Thus, the equations is of the form x² = 4ay.

Since the parabola passes through (5, 2), we have

5² = 4a(2)

25 = 8a

⇒ a =

Therefore, the equation of the parabola is

x² = 4 = ⇒ 2x² = 25y.