Subtangent and Subnormal:
Let tangent and normal at any point P (x1, y1) on parabola y2 = 4ax meet the axis at T and G respectively. Then PT is called as length of the tangent at the point P and PG is called as length of the normal at P.
NT is called as subtangent and NG the subnormal at point P. The coordinates of points T and G can be easily found by putting x = 0 in the equations of tangent and normal at P. It is evident from figure that
Subtangent = NT = twice the abscissa of P
Subnormal = NG = 2a = semilatus rectum
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Example : If P is point on the parabola y2 = 4ax so that the subtangent and subnormal at P are equal, find the coordinates of point P.
Solution: Let P (x, y) be the required point.
Length of subtangent = twice the abscisse of P = 2x
Length of subnormal = 2a
As subtangent = subnormal, 2x = 2a Þ x = a => y2 = 4a2 => y = ± 2a
∴ The required points are (a, 2a) and (a, -2a).
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