Locus-Pocus

Refer to the diagram below:

Let PQ be the variable chord, the tangent at whose extremities meet at T(h,k). Then, PQ is the chord of contact of the tangents drawn from T. As such the equation of PQ can be written as

ky = 2a(x+h) => ky - 2ax = 2ah ------------(1)

AP and AQ are the straight lines joining the points of itersection of the line (1) with the parabola y² = 4ax. As such, we can obtain the equation representing AP and AQ by homogenizing the equation of the parabola using (1). So the combined equation of AP and AQ is

y² = 4ax (ky - 2ax2ah)

which is same as

4ax² - 2kxy + hy² = 0

Since the angle between these pair of lines is tan^-1 b, we get

tan(tan^-1 b) = 2√k² - 4ah4a+h

which gives, after simplification,

4(k² - 4ah) = b²(h+4a)²

Since, this equation is satisfied by all positions of T(h,k), the locus of T becomes

4(y² - 4ax) = b²(x+4a)²

Thank you