Mathematics - Circle Question with Solution | TestHub

Equation of any circle through $(0,0)$ and $(1,0)$ $(x-0)(x-1)+(y-0)(y-0)+\lambda \left|\begin{array}{lll}x& y& 1\\ 0& 0& 1\\ 1& 0& 1\end{array}\right|=0$

$\Rightarrow x^2+y^2-x+\lambda y=0$

If it represents $C_3$, its radius $=1$

$\Rightarrow 1=(1/4)+\left({\lambda }^2/4\right)\Rightarrow \lambda =\pm \sqrt{3}$

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 $C_2$ interest and are of unit radius, their common tangents are parallel to the joining their centres $(0,0)$ and $(1/2,\sqrt{3}/2)$.

So, let the equation of a common tangents be $\sqrt{3}x-y+2=0$

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 tangent makes positive intercept on the $y$-axis and negative on the $x$-axis and hence its equation to $\sqrt{3}x-y+2=0$