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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