Standard form of Parabola x\(^{2}\) = 4ay

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 co-ordinate of the vertex is at (0, 0), the co-ordinates 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 positive y-axis, the length of its latus rectum = 4a and the distance between its vertex and focus is a.


Solved example based on the standard form of parabola x\(^{2}\) = 4ay:

Find the axis, co-ordinates of vertex and focus, length of latus rectum and the equation of directrix of the parabola x\(^{2}\) = 6y.

Solution:

The given parabola x\(^{2}\) = 6y

⇒ x\(^{2}\) = 4 ∙ \(\frac{3}{2}\) y

Compare the above equation with standard form of parabola x\(^{2}\) = 4ay, we get, a = \(\frac{3}{2}\).

Therefore, the axis of the given parabola is along positive y-axis and its equation is x = 0.

The co-ordinates of its vertex are (0, 0) and the co-ordinates of its focus are (0, 3/2); the length of its latus rectum = 4a = 4 \(\frac{3}{2}\) = 6 units and the equation of its directrix is y = -a i.e., y = -\(\frac{3}{2}\) i.e., y + \(\frac{3}{2}\) = 0 i.e., 2y + 3 = 0.






11 and 12 Grade Math 

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