A circle `C_1` of radius b touches the circle `x^2 + y^2 =a^2` externally and has its centre on the positiveX-axis; another circle `C_2` of radius c touches the circle `C_1`, externally and has its centre on the positive x-axis. Given `a lt b lt c` then three circles have a common tangent if a,b,c are in

Options

A. AP
B. GP
C. HP
D. none of these

Answer

2

Explanation

From the figure,
`(a)/(x) =(b)/(x+a+b)=(c )/(xa+2b+c)`
`:. ax+a^(2)+ab=bx` (Comparing first two )
or `x=(a^(2)+ab)/(b-a)`
`ax+a^(2)+2ab+ac=cx` (Comparing first and third )
`:. (a^(2)+ab)/(b-a)=(a^(2)+2ab+ac)/(c-a)` ,brgt `a^(2)c-a^(3)+abc-a^(2)b=a^(2)b+2ab^(2)+abc-a^(3)-2a^(2)b-a^(2)c`
or `ac=b^(2)`
Therefore, a,b, and c are in GP.