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Eccentric Angle : Ellipse

A line meets ...

Question

A line meets the co-ordinate axes in $A$ and $B$ . A circle is circumscribed about the triangle $O A B$ . If $d_{1}$ and $d_{2}$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ , respectively, then the diameter of the circle is

\frac{2 d_{1} + d_{2}}{2}

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\frac{d_{1} + 2 d_{2}}{2}

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d_{1} + d_{2}

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\frac{d_{1} d_{2}}{d_{1} + d_{2}}

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Solution

The correct option is C $d_{1} + d_{2}$
Let the circle be $x^{2} + y^{2} + 2 g x + 2 f y = 0$
Tangent at the origin is
$g x + f y = 0$
$d_{1} = \frac{2 g^{2}}{\sqrt{g^{2} + f^{2}}}$ and $d_{2} = \frac{2 f^{2}}{\sqrt{g^{2} + f^{2}}}$
$d_{1} + d_{2} = 2 \sqrt{g^{2} + f^{2}}$ $=$ diameter of the circle

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A line meets the co-ordinate axes in A and B. A circle is circumscribed about the triangle OAB. If d1 and d2 are the distances of the tangent to the circle at the origin O from the points A and B, respectively, then the diameter of the circle is

The correct option is C d1+d2Let the circle be x2+y2+2gx+2fy=0Tangent at the origin isgx+fy=0d1=2g2√g2+f2 and d2=2f2√g2+f2d1+d2=2√g2+f2 = diameter of the circle

A line meets the co-ordinate axes in $A$ and $B$ . A circle is circumscribed about the triangle $O A B$ . If $d_{1}$ and $d_{2}$ are the distances of the tangent to the circle at the origin $O$ from the points $A$ and $B$ , respectively, then the diameter of the circle is