If OA and OB are the tangents from the origin | vedclass

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(B) The center of the circle $C$ is $(-g, -f)$ and the radius $r$ is $\sqrt{g^2 + f^2 - c}$.Let $O$ be the origin $(0, 0)$. The length of the tangent

Vedclass · Accepted Answer

$\sqrt {c({g^2} + {f^2} - c)} $

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(B) The center of the circle $C$ is $(-g, -f)$ and the radius $r$ is $\sqrt{g^2 + f^2 - c}$.Let $O$ be the origin $(0, 0)$. The length of the tangent $OA$ from the origin to the circle is given by $\sqrt{S_1}$,where $S_1$ is the value of the circle equation at $(0, 0)$.Thus,$OA = \sqrt{0^2 + 0^2 + 2g(0) + 2f(0) + c} = \sqrt{c}$.In the right-angled triangle $\Delta OAC$,$\angle OAC = 90^\circ$ because the tangent is perpendicular to the radius at the point of contact.The area of $\Delta OAC = \frac{1}{2} 	imes OA 	imes AC = \frac{1}{2} 	imes \sqrt{c} 	imes \sqrt{g^2 + f^2 - c}$.The quadrilateral $OACB$ is composed of two congruent triangles,$\Delta OAC$ and $\Delta OBC$.Therefore,the area of quadrilateral $OACB = 2 	imes 	ext{Area}(\Delta OAC) = 2 	imes \frac{1}{2} 	imes \sqrt{c} 	imes \sqrt{g^2 + f^2 - c} = \sqrt{c(g^2 + f^2 - c)}$.

If $OA$ and $OB$ are the tangents from the origin to the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ and $C$ is the centre of the circle,the area of the quadrilateral $OACB$ is

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