Let `S_(1)` and `S_(2)` denote the circles `x^(2)+y^(2)+10x - 24y - 87 =0` and `x^(2) +y^(2)-10x -24y +153 = 0` respectively. The value of a for which the line `y = ax` contains the centre of a circle which touches `S_(2)` externally and `S_(1)` internally is

Options

A. `+-(3)/(10)`
B. `+-(1)/(5)`
C. `+-(sqrt(13))/(10)`
D. `+-(10)/(13)`

Answer

C

Explanation

`S_(1): C_(1) (-5,12), r_(1) = 16`
`S_(2): C_(2) (5,12), r_(2) =4`
`C_(1)C_(2) = r +4` (where C is the centre of circle touching `C_(2)` externally and `C_(1)` internally)
`C C_(1) = 16 -r` (Not `r -16, :' S_(2)` is contained by `S_(1))`
`rArr C C_(1) + C C_(2) = 20`
`:.` Locus of C is the ellipse with foci at `C_(1)` and `C_(2)` and length of major axis = 20
`:.` Locus of C is
`(x^(2))/(100) = ((y-12)^(2))/(75) =1` (i)
According to question `y = ax` is tangent to the ellipse from (0,0). Equation of tangent having slope m to (i) is `y - 12 = mx +- sqrt(100 m^(2) + 75)`
But this is passing through (0,0)
`rArr - 12 =- sqrt(100 m^(2) + 75)`
`rArr m^(2) = (69)/(100)`
`rArr a = +- (sqrt(13))/(10)`