(C) The length of the tangent from a point $(h, k)$ to a circle $x^2+y^2+2gx+2fy+c=0$ is given by $\sqrt{h^2+k^2+2gh+2fk+c}$.For the circle $x^2+y^2-1

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(C) The length of the tangent from a point $(h, k)$ to a circle $x^2+y^2+2gx+2fy+c=0$ is given by $\sqrt{h^2+k^2+2gh+2fk+c}$.For the circle $x^2+y^2-16=0$,the length of the tangent is $L_1 = \sqrt{h^2+k^2-16}$.For the circle $x^2+y^2+2x+2y=0$,the length of the tangent is $L_2 = \sqrt{h^2+k^2+2h+2k}$.Given that $L_1 = 2L_2$,we have $\sqrt{h^2+k^2-16} = 2\sqrt{h^2+k^2+2h+2k}$.Squaring both sides,we get $h^2+k^2-16 = 4(h^2+k^2+2h+2k)$.$h^2+k^2-16 = 4h^2+4k^2+8h+8k$.Rearranging the terms,we get $3h^2+3k^2+8h+8k+16=0$.

Class 11 Mathematics 1 Conic Sections: Circle Mix Examples-Circle and System of Circles

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If the length of the tangent from $(h, k)$ to the circle $x^2+y^2=16$ is twice the length of the tangent from the same point to the circle $x^2+y^2+2x+2y=0$,then:

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A
$h^2+k^2+4h+4k+16=0$
B
$h^2+k^2+3h+3k=0$
C
$3h^2+3k^2+8h+8k+16=0$
D
$3h^2+3k^2+4h+4k+16=0$

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Mix Examples-Circle and System of Circles

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If the length of the tangent from $(h, k)$ to the circle $x^2+y^2=16$ is twice the length of the tangent from the same point to the circle $x^2+y^2+2x+2y=0$,then:

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Suppose $O(0,0)$ is the origin and the line $L = x + y - \lambda = 0$ meets the curve $x^2 + y^2 - 2x - 4y + 2 = 0$ at $A$ and $B$. If $\angle AOB = 90^{\circ}$,then the distance between such lines $L = 0$ is

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