Question

$C_{1}and{C}_2$ 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

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Equation of any circle through (0,0) and
$(1,0)(x-0)(x-1)+(y-0)(y-0)+\lambda \left|\begin{array}{ccc}x& y& 1\\ 0& 0& 1\\ 1& 0& 1\end{array}\right|=0\Rightarrow x^2+y^2-x+\lambda y=0$
if it represents $C_3$ , its radius = 1 $\Rightarrow 1=\frac{1}{4}+\frac{{\lambda }^2}{4}\Rightarrow \lambda =\pm \sqrt{3}$

As the centre of $C_3$ , lies above the x-axis, we take $\lambda =-\sqrt{3}$ and thus an equation of $C_3$ is $x^2+y^2-x-\sqrt{3}y=0.$ Since $C_1$ and $C_3$ intersect and are of unit radius, their common tangents are parallel to the joining their centres (0,0) and $(\frac{1}{2},\frac{\sqrt{3}}{2}).$So, let the equation of a common tangents be $\sqrt{3}x-y+k=0$ it will touch $C_1$, if $\left|\frac{k}{\sqrt{3+1}}\right|=1\Rightarrow k=\pm 2$ From the figure, we observe that the required tangents makes positive intercept on the y-axis and negative on the x-axis and hence its equation is $\sqrt{3}x-y+2=0$