If a triangle is formed by any three tangents of a parabola Y2 = 4ax, two of whose vertices lie on the parabola x2 = 4by, then find the locus of the third vertex.

Question

Vinti Singhai · Answer

Let three points at which tangents are drawn be (at1^2, 2at1);(at2^2, 2at2);(at3^2, 2at3);Hence vertices of triangle will be:at1t2,a(t1 + t2);at2t3,a(t2 + t3);at1t3,a(t1 + t3)Let at2t3,a(t2 + t3);at1t3,a(t1 + t3) lie on parabola x^2=4byhence,a^2t2^2t3^2 = 4b (t2 + t3) or at2^2t3^2 = 4b(t2 + t3)........................1Similarly,at1^2t3^2 = 4b(t1 + t3)............................................2Subtract 1 and 2 & divide 1 and 2 to get relation b/w t1t2 and t1+t2.......................3Locus:h=at1t2 and k= a(t1+t2). Use 3 to get the final answer.Vinti SinghaiaskIITians Faculty,B.E(Honors) in Elec &Comm, Gold Medalist

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If a triangle is formed by any three tangents of a parabola Y2 = 4ax, two of whose vertices lie on the parabola x2 = 4by, then find the locus of the third vertex.

If a triangle is formed by any three tangents of a parabola Y2 = 4ax, two of whose vertices lie on the parabola x2 = 4by, then find the locus of the third vertex.

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