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Number of Common Tangents to Two Circles in Different Conditions

The number of...

Question

The number of common tangents to the following pairs of circles $x^{2} + y^{2} = 4, x^{2} + y^{2} - 6 x - 8 y + 16 = 0$ is

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Solution

$C_{1} (0, 0) & r_{1} = 2$

$C_{2} (3, 4) & r_{2} = \sqrt{9 + 16 - 16} = \sqrt{9} = 3$

$C_{1} C_{2} = \sqrt{9 + 16} = \sqrt{25} = 5$

$r_{1} + r_{2} = 3 + 2 = 5 \Rightarrow C_{1} C_{2} = r_{1} + r_{2}$

The given two circles touch each other externally.
Number of common tangents $= 3$

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Number of Common Tangents to Two Circles in Different Conditions

Standard XII Mathematics

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