The locus of the point of intersection of two tangents to the parabola y²=4 a x, which are at right angle to one another is

Step-by-step solution

Let the two tangents to the parabola (y²=4 a x ) be (P T ) and (Q T ) which are at right angle to one another at ( T ( h , k ) ). Then we have a find the locus of ( T ( h , k ) ). We know that (y=m x+ a m ), where (m ) is the slope is the equation of tangent to the parabola ( y ²=4 ax ) for all ( m ). Since this tangent to the parabola will pass through ( T ( h , k ) ), so ( k = mh + a m ; or m ² ~h - mk + a =0 ) This is a quadratic equation in (m ), so will have two roots, say (m₁ ) and (m₂ ), then ( m ₁+ m ₂= k h