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  • A line meets the co-ordinate axes in . A circle is circumscribed about the triangle OAB.

A line meets the co-ordinate axes in A\, \&\, B. A circle is circumscribed about the triangle OAB. If \mathrm{d}_{1} \& \mathrm{~d}_{2} are the distances of the tangent to the circle at the origin \mathrm{O}  from the points \mathrm{A} and \mathrm{B}  respectively, the diameter of the circle is :

Option: 1

\frac{2 \mathrm{~d}_{1}+\mathrm{d}_{2}}{2}


Option: 2

\frac{\mathrm{d}_{1}+2 \mathrm{~d}_{2}}{2}


Option: 3

\mathrm{d_{1}+d_{2}}


Option: 4

\frac{\mathrm{d}_{1} \mathrm{~d}_{2}}{\mathrm{~d}_{1}+\mathrm{d}_{2}}


Answers (1)

best_answer

 Let the circle be \mathrm{x^{2}+y^{2}+2 g x+2 f y=0}

Tangent at the origin is \mathrm{\mathrm{gx}+\mathrm{fy}=0}

\mathrm{\mathrm{d}_{1}=\frac{2 \mathrm{~g}^{2}}{\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}}}}
\mathrm{\mathrm{d}_{2}=\frac{2 \mathrm{f}^{2}}{\sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}}}}

\mathrm{\Rightarrow \quad \mathrm{d}_{1} \mathrm{~d}_{2}=2 \sqrt{\mathrm{g}^{2}+\mathrm{f}^{2}}=\text{diameter of the circle}}

Posted by

Ritika Harsh

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