First slide

Definition of a circle

Question

Two circles are externally tangent. Line s PAB and PA'B' are common tangents with A and A' on the smaller circle and B and B' the on the larger circle. If PA= AB=4, then the square of the radius of the circle is

Moderate

Solution

$\frac{{AC}_{1}}{PA} = \frac{{BC}_{2}}{PB}$
or ${BC}_{2} = 2 {AC}_{1}$
${PC}_{1} = \sqrt{16 + r^{2}}$
and ${PC}_{2} = \sqrt{64 + 4 r^{2}} = 2 \sqrt{16 + r^{2}}$
${PC}_{2} - {PC}_{1} = 3 r$
or $2 \sqrt{16 + r^{2}} - \sqrt{16 + r^{2}} = 3 r$
or $\sqrt{16 + r^{2}} = 3 r$
or $16 + r^{2} = 9 r^{2} or r^{2} = 2$

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