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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 liney = ax contains the centre of a circle which touches S_(2) externally and S_(1) internally is |
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Answer» `+-(3)/(10)` `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)` |
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