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

Question

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 (A) 12 (B) 10 (C) 14 (D) 18

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="Solution" src="https://cdn.tardigrade.in/q/nta/m-2lfuddrdp3me.png" /> From the figure, 9r2+(36)2=(5 r + 14)2 ⇒ 4 r2 + 35 r - 25 ⋅ 11 = 0 ⇒ (r - 5)(4 r + 55)=0 Hence, r=5cms⇒ 2r=10cms

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

$12$

$10$

$14$

$18$

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

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

12

10

14

18