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Director Circle

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Question

The equation of a circle $C_{1}$ is $x^{2} + y^{2} = 4$ . The locus of the intersection of othogonal tangents to the circle is the curve $C_{2}$ and the locus of the intersection of perpendicular tangents to the curve $C_{2}$ is the curve $C_{3}$ . Then

A

C_{3}

is a circle

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B

The area enclosed by the curve

C_{3}

is

8 π

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C

C_{2}

and

C_{3}

are circles with the same centre

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D

None of these

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Solution

The correct options are
A $C_{2}$ and $C_{3}$ are circles with the same centre
D $C_{3}$ is a circle
The locus of point of intersection of perpendicular tangents to the circle $C_{1} : x^{2} + y^{2} = 4$ is a directer circle
which is $C_{2} : x^{2} + y^{2} = 2 (4) = 8$
and the locus of point of intersection of perpendicular tangents to the circle $C_{2} : x^{2} + y^{2} = 8$ is a directer circle
which is $C_{3} : x^{2} + y^{2} = 2 (8) = 16$
and $C_{2}$ & $C_{3}$ have same center.
Ans: C

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