11.0 Common Tangents

  Circles

(a) When two circles touch each other externally:

• Distance between their centres ${C_1}$ and ${C_2}$$=$ Sum of their radii $$\left| {{C_1}{C_2}} \right| = {r_1} + {r_2}$$
• Number of common tangents$=3$.
• ${C_1} \equiv ({x_1},{y_1})$, ${C_2} \equiv ({x_2},{y_2})$, $\frac{{{C_1}P}}{{{C_2}P}} = \frac{{{r_1}}}{{{r_2}}}$ then, coordinates of $P$ is $$\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)$$

(b) When two circles touch each other internally:

• $\left| {{C_1}{C_2}} \right| = \left| {{r_1} - {r_2}} \right|$
• Number of common tangents$=1$.
• ${C_1} \equiv ({x_1},{y_1})$, ${C_2} \equiv ({x_2},{y_2})$, $\frac{{{C_1}P}}{{{C_2}P}} = \frac{{{r_1}}}{{{r_2}}}$ then, coordinates of $P$ is $$\left( {\frac{{{r_1}{x_2} - {r_2}{x_1}}}{{{r_1} - {r_2}}},\frac{{{r_1}{y_2} - {r_2}{y_1}}}{{{r_1} - {r_2}}}} \right)$$

(c) When two circles do not intersect each other:

• $\left| {{C_1}{C_2}} \right| > {r_1} + {r_2}$
• Number of common tangents$=4$.

(d) When two circles intersect each other:

• $\left| {{r_1} - {r_2}} \right| < $Distance between ${C_1}$ and ${C_2}$$ < {r_1} + {r_2}$
• Number of common tangents$=2$.