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Question

The tangents drawn from a point P to the ellipse make angles θ1 and θ2 with the major axis; find the locus of P when tanθ1tanθ2 is constant (=d).

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Solution

x2a2+y2b2=1

Let point P be (h,k)

Equation of tangent in slope form is y=mx±a2m2+b2

It passes through (h,k)

k=mh±a2m2+b2kmh=±a2m2+b2

Squaring both sides

k2+m2h22hkm=a2m2+b2(h2a2)m22hkm+k2b2=0m1+m2=ba=2hkh2a2........(i)m1m2=ca=k2b2h2a2.........(ii)

(tanθ1tanθ2)2=(tanθ1+tanθ2)24tanθ1tanθ2d2=(m1+m2)24m1m2

Substituting (i) and (ii)

d2=(2hkh2a2)24k2b2h2a2d2=4h2k24(h2a2)(k2b2)(h2a2)2d2(h2a2)2=4(h2k2h2k2+a2k2+h2b2a2b2)d2(h2a2)2=4(a2k2+h2b2a2b2)

Replacing h by x and k by y

d2(x2a2)2=4(a2y2+b2x2a2b2)

is the required equation of locus.


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