We will discuss about the standard form of parabola x\(^{2}\) = 4ay
Equation y\(^{2}\) = 4ax (a > 0) represents the equation of a parabola whose coordinate of the vertex is at (0, 0), the coordinates of the focus are (0, a), the equation of directrix is y = a or y  a = 0, the equation of the axis is x = 0, the axis is along negative yaxis, the length of its latus rectum = 4a and the distance between its vertex and focus is a.
Solved examples based on the standard form of parabola x\(^{2}\) = 4ay:
1. Find the axis, coordinates of vertex and focus, length of latus rectum and the equation of directrix of the parabola x\(^{2}\) = 16y
Solution:
The given parabola x\(^{2}\) = 16y
⇒ x\(^{2}\) = 4 ∙ 4 y
Compare the above equation with standard form of parabola x\(^{2}\) = 4ay, we get, a = 4.
Therefore, the axis of the given parabola is along negative yaxis and its equation is x = 0
The coordinates of its vertex are (0, 0) and the coordinates of its focus are (0, 4); the length of its latus rectum = 4a = 4 ∙ 4 = 16 units and the equation of its directrix is y = a i.e., y = 4 i.e., y  4 = 0.
2. Find the axis, coordinates of vertex and focus, length of latus rectum and the equation of directrix of the parabola 3x\(^{2}\) = 8y
Solution:
The given parabola 3x\(^{2}\) = 8y
⇒ x\(^{2}\) = \(\frac{8}{3}\)y
⇒ x\(^{2}\) = 4 ∙ \(\frac{2}{3}\) y
Compare the above equation with standard form of parabola x\(^{2}\) = 4ay, we get, a = \(\frac{2}{3}\).
Therefore, the axis of the given parabola is along negative yaxis and its equation is x = 0
The coordinates of its vertex are (0, 0) and the coordinates of its focus are (0, \(\frac{2}{3}\)); the length of its latus rectum = 4a = 4 ∙ \(\frac{2}{3}\) = \(\frac{8}{3}\) units and the equation of its directrix is y = \(\frac{2}{3}\) i.e., 3y = 2 i.e., 3y  2 = 0.
`11 and 12 Grade Math
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