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