The range of values of `lambda` for which the circles `x^(2)+y^(2)=4` and `x^(2)+y^(2)-4lambda x + 9 = 0` have two common tangents, is

Options

A. [-13/8, 13/8]
B. `(-oo, -13//8) uu(13//8, oo)`
C. `(1, 13//8)`
D. none of these

Answer

B

Explanation

The coordinates of the centres and radii of the two circles are:
Centres: `C_(1)(0, 1) " " C_(2)(2lambda, 0)`
Radii: `" " r_(1)=2 " " r_(2)=sqrt(4lambda^(2)-9), lambda in (-oo, -3//2) uu (3//2, oo)`
The two circles will have exactly two common tangents , if
`|r_(1)-r_(2)| lt C_(1)C_(2) lt r_(1) + r_(2)`
`rArr | 2- sqrt(4lambda^(2)-9)| lt | 2 lambda| lt 2 + sqrt(4lambda^(2)-9)`
`rArr |2 lambda| lt 2 + sqrt(4lambda^(2)-9)[:'|2-sqrt(4lambda^(2)-9)| lt | 2 lambda| ` for all `lambda in R]`
`rArr | 2 lambda|-2 lt sqrt(4lambda^(2)-9)`
`rArr (|2lambda|-2)^(2) lt 4lambda^(2)-9 " " [:. | 2 lambda | - 2 gt 0 ]`
`rArr -8 |lambda| lt -13`
`rArr | lambda| gt (13)/(8) rArr lambda in (-oo, -13//8) uu (13//8, oo)`