The set of all real values of \lambda for which exactly two common tangents can be drawn to the circles

x^{2}+y^{2}-4x-4y+6=0  and x^{2}+y^{2}-10x-10y+\lambda =0  is the interval :

  • Option 1)

     (12, 32)

  • Option 2)

     (18, 42)

  • Option 3)

     (12, 24)

  • Option 4)

     (18, 48)

 

Answers (1)

As we learnt in

Common tangents of two circle -

When they intersect, there are two common tangents, both of them being direct.

- wherein

 

 S_{1}:x^{2}+y^{2}-4x-4y+6=0

r_{1}\:=\:\sqrt{2^{2}+2^{2}-6}\:=\sqrt{2}

C_{1}:\:(2,2)

S_{2}:\:x^{2}+y^{2}-10x-10y+\lambda=0

r_{2}\:=\sqrt{5^{2}+5^{2}-\lambda}\:=\:\sqrt{50-\lambda}

C_{2}:\:(5,5)

For two tangents

r_{1}-r_{2}< C_{1}C_{2}< \:r_{1}+r_{2}

\sqrt{50-\lambda}-\sqrt{2}< 3\sqrt{2}< \sqrt{2}+\sqrt{50-\lambda}

We get \sqrt{50-\lambda}< 4\sqrt{2}\; \Rightarrow \lambda> 18

\sqrt{50-\lambda}> 2\sqrt{2}\; \Rightarrow \lambda< 42

\lambda\epsilon (18,42)


Option 1)

 (12, 32)

This option is incorrect.

Option 2)

 (18, 42)

This option is correct.

Option 3)

 (12, 24)

This option is incorrect.

Option 4)

 (18, 48)

This option is incorrect.

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