Two tangents to the circle x^{2}+y^{2}=4 at t | vedclass

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(A) The circle is $x^{2}+y^{2}=4$,so its radius $OA = 2$ and the center $O$ is $(0,0)$.Since $MA$ is a tangent,$\angle OAM = 90^{\circ}$.In the right-

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$4 \sqrt{3}$ sq. units

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(A) The circle is $x^{2}+y^{2}=4$,so its radius $OA = 2$ and the center $O$ is $(0,0)$.Since $MA$ is a tangent,$\angle OAM = 90^{\circ}$.In the right-angled triangle $	riangle OAM$,the hypotenuse $OM = 4$ and the side $OA = 2$.Using the Pythagorean theorem,$MA = \sqrt{OM^{2} - OA^{2}} = \sqrt{4^{2} - 2^{2}} = \sqrt{16 - 4} = \sqrt{12} = 2\sqrt{3}$.The quadrilateral $MAOB$ consists of two congruent right-angled triangles,$	riangle OAM$ and $	riangle OBM$.Therefore,the area of quadrilateral $MAOB = 2 	imes 	ext{Area}(	riangle OAM)$.Area $(	riangle OAM) = \frac{1}{2} 	imes OA 	imes MA = \frac{1}{2} 	imes 2 	imes 2\sqrt{3} = 2\sqrt{3}$.Thus,the area of quadrilateral $MAOB = 2 	imes 2\sqrt{3} = 4\sqrt{3}$ sq. units.

Two tangents to the circle $x^{2}+y^{2}=4$ at the points $A$ and $B$ meet at $M(-4,0)$. The area of the quadrilateral $MAOB$,where $O$ is the origin,is

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