Let S,S′ be the foci of the ellipse x2a2+y2b2=1 whose eccentricity is 'e'. P is a variable point on the ellipse. Consider the locus of the incentre of the △PSS′
The locus of the incentre of the △PSS′ is
A
ellipse
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B
hyperbola
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C
parabola
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D
circle
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Solution
The correct option is A ellipse
Let the coordinates of P be (acosθ,bsinθ) Here, SP= Focal distance of point P So, SP=a−aecosθ S′P=a+aecosθ SS′=2ae
If (h,k) are the coordinates of the incenter of ΔPSS′, then h=2ae(acosθ)+a(1−ecosθ)(−ae)+a(1+ecosθ)ae2ae+a(1−ecosθ)+a(1+ecosθ)=aecosθ⋯(i) and k=2ae(bsinθ)+a(1−ecosθ)×0+a(1+ecosθ)×02ae+a(1−ecosθ)+a(1+ecosθ)=ebsinθ(e+1)⋯(ii) From (i) and (ii), hae=cosθk(e+1)eb=sinθ Eliminating θ, we get x2a2e2+y2(be(e+1))2=1 Which clearly represents an ellipse.