    Question

# A variable straight line through A(−1,−1) is drawn to cut the circle x2+y2=1 at the points B,C. If P is chosen on the line ABC such that AB,AP,AC are in H.P then the locus of P is

A
x+y+1=0
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B
x+y1=0
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C
xy+1=0
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D
xy1=0
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Solution

## The correct option is A x+y+1=0Let locus at point P=(h,k),AP=r, inclination of line ABC be θ then any point on the line will be of form (x,y)=(−1+rcosθ,−1+rsinθ) For this point to lie on the circle x2+y2=1 ⇒(−1+rcosθ)2+(−1+rsinθ)2=1 ⇒r2−2(cosθ+sinθ)r+1=0 r1,r2 are its roots Let r1=AB,r2=AC If AB,AP,AC are in H.P ⇒AP=2AB.ACAB+AC ⇒r=2r1⋅r2r1+r2 ⇒r=22(cosθ+sinθ) ⇒rcosθ+rsinθ=1 ⇒(h+1)+(k+1)=1 ⇒h+k+1=0 so, locus is x+y+1=0  Suggest Corrections  0      Similar questions  Related Videos   MATHEMATICS
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