Question : The distance between the centres of two circles having radii of 24 cm and 18 cm, respectively, is 48 cm. Find the length (in cm) of a direct common tangent to the two circles.
Option 1: $20 \sqrt{6}$
Option 2: $18 \sqrt{7}$
Option 3: $45$
Option 4: $22 \sqrt{5}$
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Correct Answer: $18 \sqrt{7}$
Solution :
Given: Distance between centres of two circles = 48 cm
The radius of the two circles = 24 cm and 18 cm.
The length of direct common tangent = $\sqrt{D^{2}-(R_1-R_2)^{2}}$ to find the desired value.
Here $D$ is the distance between the centres and $R_1, R_2$ are the radii of two circles.
Putting the values, we have:
Length of direct common tangent = $\sqrt{48^{2}–(24–18)^{2}}$
= $\sqrt{48^{2}–6^{2}}$
= $\sqrt{(48–6)(48+6)}$
= $\sqrt{42×54}$
= $18\sqrt{7}$ cm
Hence, the correct answer is $18\sqrt{7}$.
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