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COM of Center of Masses

C1:x2+y2+2x-3...

Question

$C_{1} : x^{2} + y^{2} + 2 x - 3 = 0, C_{2} : x^{2} + y^{2} - 2 x - 3 = 0, C_{3} : x^{2} + y^{2} - 2 y - 3 = 0,$ are three circles

A

C_{1}

passes through the centre of

C_{2}

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B

C_{2}

passes through the centre of

C_{1}

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C

centres of the three circles form an equilateral triangle

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D

sum of the radii of

C_{1}

and

C_{2}

is twice the radius of

C_{3}

.

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Solution

The correct options are
A $C_{1}$ passes through the centre of $C_{2}$
C $C_{2}$ passes through the centre of $C_{1}$
D sum of the radii of $C_{1}$ and $C_{2}$ is twice the radius of $C_{3}$ .
Centers of $C_{1}$ , $C_{2}$ and $C_{3}$ are $(- 1, 0)$ , $(1, 0)$ and $(0, 1)$ respectively.
Since, $(1, 0)$ satisfies $C_{1}$
Therefore, $C_{1}$ passes through the centre of $C_{2}$
and $(- 1, 0)$ satisfies $C_{2}$
Therefore, $C_{2}$ passes through the centre of $C_{1}$
now, $r_{1} = \sqrt{1 + 3} = 2$
Similarly, $r_{2} = r_{3} = 2$
Therefore, $r_{1} + r_{2} = 2 r_{3}$

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