Two non congruent circles are externally tangent. The product of their radii is an integer k between 1 and 100 inclusive. Number of values of k for which the length of an external tangent is also an integer, is

Question

Two non congruent circles are externally tangent. The product of their radii is an integer k between 1 and 100 inclusive. Number of values of k for which the length of an external tangent is also an integer, is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

L2=√d2-(r2-r1)2 L=√(r1+r2)2-(r2-r1)2 <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/8af90be243a11c225a3ba9dd5650e235-.png" /> L=√4 r1 r2=2 √r1 r2 L=2 √k k ∈[1,100] L will be an integer if k=1,4,9,16,25,36,49,64,81,100 i.e. 10 values

Q. Two non congruent circles are externally tangent. The product of their radii is an integer $k$ between $1$ and $100$ inclusive. Number of values of $k$ for which the length of an external tangent is also an integer, is

$L^{2} = d^{2} - (r_{2} - r_{1})^{2}$
$L = (r_{1} + r_{2})^{2} - (r_{2} - r_{1})^{2}$

$L = 4 r_{1} r_{2} = 2 r_{1} r_{2}$
$L = 2 k k \in [1, 100]$
$L$ will be an integer if
$k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100$
i.e. $10$ values

Q. Two non congruent circles are externally tangent. The product of their radii is an integer k between 1 and 100 inclusive. Number of values of k for which the length of an external tangent is also an integer, is

L2=d2−(r2​−r1​)2​ L=(r1​+r2​)2−(r2​−r1​)2​ L=4r1​r2​​=2r1​r2​​ L=2k​k∈[1,100] L will be an integer if k=1,4,9,16,25,36,49,64,81,100 i.e. 10 values

Q. Two non congruent circles are externally tangent. The product of their radii is an integer $k$ between $1$ and $100$ inclusive. Number of values of $k$ for which the length of an external tangent is also an integer, is

$L^{2} = d^{2} - (r_{2} - r_{1})^{2}$
$L = (r_{1} + r_{2})^{2} - (r_{2} - r_{1})^{2}$

$L = 4 r_{1} r_{2} = 2 r_{1} r_{2}$
$L = 2 k k \in [1, 100]$
$L$ will be an integer if
$k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100$
i.e. $10$ values