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A chord of the circle \mathrm{x^2+y^2-a x-b y=0}, drawn from the point (a, b), is divided by the x-axis in the ratio 2: 1. Then \mathrm{a^2-3 b^2}

Option: 1

\mathrm{=0}


Option: 2

<0


Option: 3

>0


Option: 4

\neq 0


Answers (1)

best_answer

The given point $A(a, b)$ lies on the given circle. Any point on the circle is P

\mathrm{\left(\frac{a}{2}+\sqrt{\frac{a^2}{4}+\frac{b^2}{4}} \cos \theta, \frac{b}{2}+\sqrt{\frac{a^2}{4}+\frac{b^2}{4}} \sin \theta\right).}

The x-axis cuts the chord AP in the ratio 2:1.

Hence \mathrm{\frac{b+b+2 \sqrt{\frac{a^2}{4}+\frac{b^2}{4}} \sin \theta}{3}=0 \Rightarrow \sin \theta=-\frac{2 b}{\sqrt{a^2+b^2}}}

\mathrm{ \Rightarrow \quad|\sin \theta|=\left|\frac{2 b}{\sqrt{a^2+b^2}}\right|<1 \Rightarrow \quad 4 b^2<a^2+b^2 \Rightarrow a^2>3 b^2 . }

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