JEE Main & Advanced Mathematics Circle and System of Circles Common Tangents to Two Circles

Different cases of intersection of two circles :

Let the two circles be \[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}=r_{1}^{2}\]       …..(i)

and  \[{{(x-{{x}_{2}})}^{2}}+{{(y-{{y}_{2}})}^{2}}=r_{2}^{2}\]                                   …..(ii)

with centres \[{{C}_{1}}\,({{x}_{1}},\,{{y}_{1}})\] and \[{{C}_{2}}\,({{x}_{2}},\,{{y}_{2}})\] and radii \[{{r}_{1}}\] and \[{{r}_{2}}\] respectively. Then following cases may arise :

Case I : When \[|\,{{C}_{1}}{{C}_{2}}\,|\,>\,{{r}_{1}}+{{r}_{2}}\] i.e., the distance between the centres is greater than the sum of radii.

In this case four common tangents can be drawn to the two circles, in which two are direct common tangents and the other two are transverse common tangents.

The points P, T of intersection of direct common tangents and transverse common tangents respectively, always lie on the line joining the centres of the two circles and divide it externally and internally respectively in the ratio of their radii.

\[\frac{{{C}_{1}}P}{{{C}_{2}}P}=\frac{{{r}_{1}}}{{{r}_{2}}}\](externally) and \[\frac{{{C}_{1}}T}{{{C}_{2}}T}=\frac{{{r}_{1}}}{{{r}_{2}}}\](internally)

Hence, the ordinates of P and T are

\[P\equiv \left( \frac{{{r}_{1}}{{x}_{2}}-{{r}_{2}}{{x}_{1}}}{{{r}_{1}}-{{r}_{2}}},\frac{{{r}_{1}}{{y}_{2}}-{{r}_{2}}{{y}_{1}}}{{{r}_{1}}-{{r}_{2}}} \right)\]and\[T\equiv \left( \frac{{{r}_{1}}{{x}_{2}}+{{r}_{2}}{{x}_{1}}}{{{r}_{1}}+{{r}_{2}}},\frac{{{r}_{1}}{{y}_{2}}+{{r}_{2}}{{y}_{1}}}{{{r}_{1}}+{{r}_{2}}} \right)\].

Case II : When \[|\,{{C}_{1}}{{C}_{2}}\,|\,=\,{{r}_{1}}+{{r}_{2}}\] i.e., the distance between the centres is equal to the sum of radii.

In this case two direct common tangents are real and distinct while the transverse tangents are coincident.

Case III : When \[|\,{{C}_{1}}{{C}_{2}}\,|\,<\,{{r}_{1}}+{{r}_{2}}\] i.e., the distance between the centres is less than sum of radii.

In this case two direct common tangents are real and distinct while the transverse tangents are imaginary.

Case IV : When \[|\,{{C}_{1}}{{C}_{2}}\,|\,=\,\,|\,{{r}_{1}}-{{r}_{2}}\,|\,\,,\] i.e., the distance between the centres is equal to the difference of the radii.

In this case two tangents are real and coincident while the other two tangents are imaginary.

Case V : When \[|\,{{C}_{1}}{{C}_{2}}\,|\,<\,\,|\,{{r}_{1}}-{{r}_{2}}\,|\,\,,\] i.e., the distance between the centres is less than the difference of the radii.

In this case, all the four common tangents are imaginary.