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  4. Two circles have a direct common tangent with length 36cm. The shortest distance between these circles is 14cm. If the radius of the longer circle is 4 times the radius of the smaller circle, then the diameter of the smaller circle (in cms ) is

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Q. Two circles have a direct common tangent with length $36cm.$ The shortest distance between these circles is $14cm.$ If the radius of the longer circle is $4$ times the radius of the smaller circle, then the diameter of the smaller circle (in $cms$ ) is

NTA AbhyasNTA Abhyas 2020Conic Sections

Solution:

Solution
From the figure,
$9r^{2}+\left(36\right)^{2}=\left(5 r + 14\right)^{2}$
$\Rightarrow 4 r^{2} + 35 r - 25 \cdot 11 = 0$
$\Rightarrow \left(r - 5\right)\left(4 r + 55\right)=0$
Hence, $r=5cms\Rightarrow 2r=10cms$