The set of all real values of λ for which exactly two common tangents can be drawn to the circles x^2+y^2-4 x-4 y+6=0 and x^2+ y^2-10 x-10 y+λ=0 is…

The equations of the circles are $$x^2+y^2-10 x-10 y+\lambda=0$$x2+y2−10x−10y+λ=0 and $x^2+y^2-4 x-4 y+6=0$x2+y2−4x−4y+6=0 $$\begin{aligned} &C_1=\text { centre of }(1)=(5,5) \\ &\mathrm{C}_2=\text { centre of }(2)=(2,2) \\ &d=\text { distance between centres } \\ &=\mathrm{C}_1 \mathrm{C}_2=\sqrt{9+9}=\sqrt{18} \\ &r_1=\sqrt{50-\lambda}, r_2=\sqrt{2} \\ &\text { For exactly two common tangents we } \\ &\text { have } \\ &r_1-r_2 < C_1 C_2 < r_1+r_2 \\ &\Rightarrow \sqrt{50-\lambda}-\sqrt{2} < 3 \sqrt{2} < \sqrt{50-\lambda}+\sqrt{2} \\ &\Rightarrow \sqrt{50-\lambda}-\sqrt{2} < 3 \sqrt{2} \text { or } 3 \sqrt{2} < \sqrt{50-\lambda}+\sqrt{2} \\ &\Rightarrow \sqrt{50-\lambda} < 4 \sqrt{2} \text { or } 2 \sqrt{2} < \sqrt{50-\lambda} \\ &\Rightarrow 50-\lambda < 32 \text { or } 8 < 50-\lambda \\ &\Rightarrow \lambda>18 \text { or } \lambda < 42 \\ &\text { Required interval is }(18,42) \\ & \end{aligned}$$​C1​= centre of (1)=(5,5)C2​= centre of (2)=(2,2)d= distance between centres =C1​C2​=9+9​=18​r1​=50−λ​,r2​=2​ For exactly two common tangents we  have r1​−r2​<C1​C2​<r1​+r2​⇒50−λ​−2​<32​<50−λ​+2​⇒50−λ​−2​<32​ or 32​<50−λ​+2​⇒50−λ​<42​ or 22​<50−λ​⇒50−λ<32 or 8<50−λ⇒λ>18 or λ<42 Required interval is (18,42)​