Find the locus of midpoint of variable chord of parabola y2 = 4ax such that tangents make an angle Φ between them.

Dear Priyansh

let the mid point is (h,k)

and let the point where tangent touch the parabola is (at1²,2at1) and (at2²,2at2)

so 2h = a(t1² + t2²)    and  2k = 2a(t1+t2)   ...................1

from equation 1 find the value of t1t2 = (k² -2ha)/2a²

and |t1-t2| = 1/a {k² - 4ha}^1/2

so given tanΦ = |m1-m2|/(1+m1m2)

where m1 = 1/t1

            m2 =1/t2

so put these value

tanΦ =2a{k² -4ha}^1/2 / {k² -2ha}

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