In each of the, 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 vertex is (0, 0) and the parabola is symmetric about the y-axis, the equation of the parabola is either of the from x²=4ay or x² = -4ay.

The parabola passes through point (5, 2), which lies in the first quadrant.

Thus, the equation of the parabola is of the form x² = 4ay, while point (5, 2) must satisfy the equation x² = 4ay.

Thus,

5² = 4a(2)

⇒ 25 = 8a

⇒ a = 25/8

Thus, the equation of the parabola is

⇒ 2x² = 25y