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  • Let A(1,4) and B(1, -5) be two points. Let P be a point on the circle  such that <img alt="(PA)^2+(PB)^2" src

Let A(1,4) and B(1, -5) be two points. Let P be a point on the circle (x-1)^2 +(y-1)^2=1 such that (PA)^2+(PB)^2 have maximum value, then the points, P,A and B lie on:
Option: 1 a parabola
Option: 2 a straight line
Option: 3 a hyperbola
Option: 4 an ellipse

Answers (1)

best_answer

\\\text{Let P be any point on the circle }(x-1)^2+(y-1)^2=1\\\text{So, P}=(1+\cos\theta,1+\sin \theta)\\A\;(1,4),\;\;B(1,-5)\\(PA)^2+(PB)^2\\=((1+\cos\theta-1)^2+(1+\sin\theta-4)^2)+((1+\cos\theta-1)^2+(1+\sin\theta+5)^2)\\=\cos^2\theta+\sin^2\theta-6\sin\theta+9+\cos^2\theta+\sin^2\theta+12\sin\theta+36\\=47+6\sin\theta

\begin{aligned} &\text { is maximum if } \sin \theta=1\\ &\Rightarrow \sin \theta=1, \cos \theta=0\\ &\mathrm{P}(1,1) ,\mathrm{A}(1,4), \mathrm{B}(1,-5) \end{aligned}

P, A and B lies in a straight line.

Posted by

himanshu.meshram

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