Question

Prove that the area of triangle formed by the tangents frawn from x1 y1 yo parabola and chord of contact

Answer 1

Let (x1,y1) and (x2,y2) be the points of contact.

Substitute the chord ky=2a(x+h) into the parabola ya=4ax to obtain a quadratic equation in y. y1 and y2 are the roots. So we have y1+y2 and y1y2. (y1−y2)2=(y1+y2)2−4y1y2.

k(y1−y2)=2a(x1−x2) and so we can find the distance between the two points of contact.

The distance between the point (h,k) and the chord of contact is |2a(h)−k(k)+2ah4a2+k2√|.

Multiply the two distances and divide it by 2, we have the area.