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From aFrom a point O on the circle x^2 +y^2=d^2,tangent OP and OQ are drawn to the ellipse x^2/a^2+y^2/b^2=1.find locus of the midpoint of the chord PQ

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2 years ago

```							Dear Rahul Use mid point of chord of contact formula T= S1 Hope it helps RegardsArun (askIITians forum expert)
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2 years ago
```							Let any point on the given circle be O(d cos, d sin)and midpoint of chord PQ of ellipse be M(h,k).Equation of chord of contact PQ with O as an external point isT = 0 w.r.t. the ellipse x2/a2 + y2/b2 – 1 = 0.  dcosx/a2 + dsiny/b2 – 1 = 0  i.e. dcosx/a2 + dsiny/b2   =  1                          ….(1)    is the equation of PQ. Also, considering midpoint M, equation of chord PQ isT = S1 w.r.t. the ellipse x2/a2 + y2/b2 – 1 = 0. hx/a2 + ky/b2 – 1 = h2/a2 + k2/b2 – 1  i.e.hx/a2 + ky/b2   =  h2/a2 + k2/b2                 ….(2)    is the equation of PQ. Comparing (1) & (2), we geth/dcos  =  k/dsin  =  (h2/a2 + k2/b2) / 1 cos = h/d(h2/a2 + k2/b2)  &  sin = k/d(h2/a2 + k2/b2)  cos2 + sin2 = 1 [h/d(h2/a2 + k2/b2)]2 + [k/d(h2/a2 + k2/b2)]2 = 1 h2 + k2 = d2(h2/a2 + k2/b2)2 Replace h by x and k by y.   x2 + y2 = d2(x2/a2 + y2/b2)2 is the required locus.(You may simplify the above equation to get the desired result.)Pls approve.
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one year ago
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