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 positive yaxis, 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, coordinates 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 yaxis and its equation is x = 0.
The coordinates of its vertex are (0, 0) and the coordinates 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.
● The Parabola
11 and 12 Grade Math
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