$C_{1} and C_{2}$ are circles of unit radius with centres at (0,0) and (1,0) respectively. $C_{3}$ is a circle of unit radius, passes through the centres of the circles $C_{1} and C_{2}$ and have its centre above x-axis. Equation of the common tangent to $C_{1} and C_{3}$ which does not pass through $C_{2}$ is

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$x - \sqrt{3} y + 2 = 0$

3 x − y + 2 = 0

3 x + y − 2 = 0

x + 3 y + 2 = 0

answer is B.

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Detailed Solution

Equation of any circle through $(0, 0) and (1, 0)$ $(x - 0) (x - 1) + (y - 0) (y - 0)$ $+ λ |\begin{matrix} x & y & 1 \\ 0 & 0 & 1 \\ 1 & 0 & 1 \end{matrix}| = 0$ $⇒ x 2 + y 2 − x + λ y = 0$ If it represents $C 1$ , its radius $= 5 ⇒ 1 = (1 / 4) + λ 2 / 4 ⇒ λ = ± 3$
Question Image As the centre of $C 3$ , lies above the x-axis. We take $λ = − 3$ and thus an equation of $C 3$ is $x 2 + y 2 − x − 3 y = 0$ . Since $C 1 and C 2$ intersect and are of unit radius, their common tangents are parallel to the joining their centres $(0, 0)$ and $12, 32$ . So, let the equation of a common tangents be $\sqrt{3} x - y + k = 0$ it will touch $C 1$ If $|\frac{k}{\sqrt{3} + 1}| = 1 \Rightarrow K = \pm 2$
From the figure, we observe that the required tangent makes positive intercept on the x-axis and hence its equation to $3 x − y + 2 = 0$