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Question

Find the locus of a point such that the sum of its distances from (0,2) and (0,-2) is 6.

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Solution

Let p(h,k) be a point. Let the given points be A(0,2) and B(0,-2) According to the given conditin,AP+BP=6(h0)2+(k2)2+(h0)2+(k+2)2h2(k2)+h2=6h2+(k+2)2Squaring both sides,we get:h2+(k2)2=36+h2+(k+2)212h2+(k+2)2h2+k2+44k=36+h2+k2+4+4k12h2+(k+2)23h2+(k+2)2=9+2k9(h2+k2+4+4k)=81+4k2+36k (Squaring both sides)9h2+9k2+36+36k=81+4k2+36k9h2+5k245=0Hence,the locus of(h,k) is9x2+5y245=0


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