The two circles x^{2}+y^{2}=ax\; and\; x^{2}+y^{2}=c^{2}(c> 0)   touch each other if

  • Option 1)

    a=2c\;

  • Option 2)

    \; \left | a \right |=2c\;

  • Option 3)

    \; 2\left | a \right |=c\;

  • Option 4)

    \; \left | a \right |=c

 

Answers (1)

As we learnt in 

Common tangents of two circles -

When two circles touch  each other externally, there are three common tangents, two of them are direct.

 

- wherein

 

 

and

 

Common tangents of two circles -

When two circles touch each other internally, there is only one common tangent.

- wherein

 

 

S_{1}: x^{2}+y^{2}-ax=0

C_{1}: \left ( \frac{a}{2},0 \right) \: and \: r_{1}=\frac{a}{2}

S_{2}: x^{2}+y^{2}=c^{2}

C_{2}: \left ( 0,0 \right) \: and \: r_{2}=c

They can touch internally or externally

Internally,

       \left | \frac{a}{2} -c\right |=\frac{a}{2}

We get, c-\frac{a}{2}=\frac{a}{2}\Rightarrow c=a

Externally,

   \left | \frac{a}{2} +c\right |=\frac{a}{2}

\frac{-a}{2} -c=\frac{a}{2}

c=-a

So, \left | a \right |=c

 


Option 1)

a=2c\;

This option is incorrect

Option 2)

\; \left | a \right |=2c\;

This option is incorrect

Option 3)

\; 2\left | a \right |=c\;

This option is incorrect

Option 4)

\; \left | a \right |=c

This option is correct

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