Standard form of Parabola y\(^{2}\) = - 4ax

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 co-ordinate of the vertex is at (0, 0), the co-ordinates 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 x-axis; 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, co-ordinates 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 x-axis and its equation is y = 0

The co-ordinates of its vertex are (0, 0) and the co-ordinates 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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