The two circles `x^2+y^2=ax and x^2+y^2=c^2(c > 0)` touch each other if (1) `a=2c` (2) `|a|=2c` (3) `2|a|=c` (4) `|a|=c`

Options

A. c=|a|
B. 2a=|c|
C. 2c=a
D. none of these

Answer

A

Explanation

The centres of the two circles are `C_(1)(-a//2, 0) and C_(2)(0, 0)`, and there radii are `(|a|)/(2)` and c. So, the two circles will touch each other, if
`C_(1)C_(2)`=Sum or difference of radii
`rArr sqrt((-(a)/(2)-0)^(2)+(0-0)^(2))=|cpm(|a|)/(2)|`
`
`rArr (|a|)/(2)=| c pm (|a|)/(2)|`
`rArr c pm (|a|)/(2) = (|a|)/(2)`
` rArr c-(|a|)/(2)=(|a|)/(2) and c+(|a|)/(2)=(|a|)/(2)`
`rArr c=|a| or, c=0 rArr c=|a| [:' c gt 0]`