Question

If P is a point on the parabola $y ^ { 2 } = 4 a x$ such that the subtangent and subnormal at P are equal, then the coordinates of P are

• A. $( a , - 2 a )$
• B. $( 2 a , - 2 \sqrt { 2 } a )$
• C. $( 4 a , 4 a )$
• D. None of these

Answer 1

Answer: (A)

$( a , - 2 a )$

Answer 2

Since the length of the subtangent at a point on the parabola is twice the abscissa of the point and the length of the subnormal is equal to semi-latus-rectum. Therefore if $2 x = 2 a \Rightarrow x = a$

Now (x, y) lies on the parabola $y ^ { 2 } = 4 a x$ ⇒ $4 a ^ { 2 } = y ^ { 2 }$

⇒$y = \pm 2 a$

Thus the required points are $( a , 2 a )$ and $( a , - 2 a )$