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Find the locus of midpoint of variable chord of parabola y 2 = 4ax such that tangents make an angle Φ between them.

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

Grade:

1 Answers

Badiuddin askIITians.ismu Expert
147 Points
11 years ago

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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Badiuddin

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