Question : Find the distance (in cm) between the centres of two circles having radii of 32 cm and 24 cm, respectively if the length of a direct common tangent to the two circles is given as 24 cm.
Option 1: 60
Option 2: 65
Option 3: 64
Option 4: 63
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Correct Answer: 64
Solution : Direct common tangent to two circles = $\sqrt{d^{2}-(r_{1}-r_{2})^{2}}$ Let $r_{1}$ and $r_{2}$ be 32 cm and 24 cm respectively. Length of a direct common tangent to the two circles, $d = 24 \sqrt{7}$ cm Direct common tangent to two circles = $\sqrt{d^{2}-(r_{1}-r_{2})^{2}}$ ⇒ $24 \sqrt{7} = \sqrt{d^{2}-(32-24)^{2}}$ ⇒ $24 \sqrt{7} = \sqrt{d^{2}-(8)^{2}}$ ⇒ $24 \sqrt{7} = \sqrt{d^{2}-64}$ Squaring both sides, ⇒ $4032=d^{2}-64$ ⇒ $d^{2} = 4032+64$ ⇒ $d^{2} = 4096$ ⇒ $d = 64$ cm Hence, the correct answer is 64.
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