Mathematics - Circle Question with Solution | TestHub

The equations of the circles are $$x^2+y^2-10x-10y+\lambda =0$$ and $x^2+y^2-4x-4y+6=0$ $$\begin{array}{rl}& C_1=\text{ centre of }(1)=(5,5)\\ & {\mathrm{C}}_2=\text{ centre of }(2)=(2,2)\\ & d=\text{ distance between centres }\\ & ={\mathrm{C}}_1{\mathrm{C}}_2=\sqrt{9+9}=\sqrt{18}\\ & r_1=\sqrt{50-\lambda },r_2=\sqrt{2}\\ & \text{ For exactly two common tangents we }\\ & \text{ have }\\ & r_1-r_2<C_1{C}_2<r_1+r_2\\ & \Rightarrow \sqrt{50-\lambda }-\sqrt{2}<3\sqrt{2}<\sqrt{50-\lambda }+\sqrt{2}\\ & \Rightarrow \sqrt{50-\lambda }-\sqrt{2}<3\sqrt{2}\text{ or }3\sqrt{2}<\sqrt{50-\lambda }+\sqrt{2}\\ & \Rightarrow \sqrt{50-\lambda }<4\sqrt{2}\text{ or }2\sqrt{2}<\sqrt{50-\lambda }\\ & \Rightarrow 50-\lambda <32\text{ or }8<50-\lambda \\ & \Rightarrow \lambda >18\text{ or }\lambda <42\\ & \text{ Required interval is }(18,42)\\ & \end{array}$$