Three circles of radii a, b, c(a

Question

Three circles of radii a, b, c(a < b < c)touch each other externally. If they have x-axis as a common tangent, then : (A) (1/√a) = (1/√b) + (1/√c)  (B) a, b, c are in A. P (C) √a , √b , √c are in A.P (D) (1/√b) = (1/√a) + (1/√c)

Tardigrade Admin · Accepted Answer

AB =AC +CB √(b+c)2 -(b-c)2 = √(b+a)2 -(b-a)2 + √(a+c)2-(a-c)2 √bc = √ab + √ac (1/√a) = (1/√c) + (1/√b)

Q. Three circles of radii $a, b, c (a < b < c)$ touch each other externally. If they have x-axis as a common tangent, then :

$\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$

a, b, c are in A. P

$a, b, c$ are in A.P

$\frac{1}{b} = \frac{1}{a} + \frac{1}{c}$

$A B = A C + CB$
$(b + c)^{2} - (b - c)^{2}$
$= (b + a)^{2} - (b - a)^{2} + (a + c)^{2} - (a - c)^{2}$
$b c = ab + a c$
$\frac{1}{a} = \frac{1}{c} + \frac{1}{b}$

Q. Three circles of radii a,b,c(a<b<c)touch each other externally. If they have x-axis as a common tangent, then :

a​1​=b​1​+c​1​

a, b, c are in A. P

a​,b​,c​ are in A.P

b​1​=a​1​+c​1​

AB=AC+CB (b+c)2−(b−c)2​ =(b+a)2−(b−a)2​+(a+c)2−(a−c)2​ bc​=ab​+ac​ a​1​=c​1​+b​1​

Q. Three circles of radii $a, b, c (a < b < c)$ touch each other externally. If they have x-axis as a common tangent, then :

$\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$

$a, b, c$ are in A.P

$\frac{1}{b} = \frac{1}{a} + \frac{1}{c}$

$A B = A C + CB$
$(b + c)^{2} - (b - c)^{2}$
$= (b + a)^{2} - (b - a)^{2} + (a + c)^{2} - (a - c)^{2}$
$b c = ab + a c$
$\frac{1}{a} = \frac{1}{c} + \frac{1}{b}$