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  • Find the range of parameter ' a' for which the variable line   lies between the cir

Find the range of parameter ' a' for which the variable line y=2 x+a  lies between the circles x^2+y^2-2 x-2 y+1=0 and  x^2+y^2-16 x-2 y+61=0  without intersecting or touching either circle.

Option: 1

(2 \sqrt{5}-15, \sqrt{5}-1)


Option: 2

(\sqrt{5}-15,-\sqrt{5}-1)


Option: 3

(2 \sqrt{5}-15,-\sqrt{5}-1)


Option: 4

 None the these


Answers (1)

best_answer

The given circles are \mathrm{C_1:(x-1)^2+(y-1)^2=1\, and\, C_2:(x-8)^2+(y-1)^2=4 }
The line  \mathrm{y-2 x-a=0 } will lie between these circle if center of the circles lie on opposite sides of the line, i.e., \mathrm{(1-2-a)(1-16-a)<0 \Rightarrow a \in(-15,-1) }
Line wouldn't touch or intersect the circles if, \mathrm{\frac{|1-2-\mathrm{a}|}{\sqrt{5}}>1, \frac{|1-16-\mathrm{a}|}{\sqrt{5}}>2 }
\mathrm{\begin{aligned} & \Rightarrow \quad|1+a|>\sqrt{5}, \quad|15+a|>2 \sqrt{5} \\ & \Rightarrow \quad a>\sqrt{5}-1 \text { or } a<-\sqrt{5}-1, a>2 \sqrt{5}-15 \text { or } a<-2 \sqrt{5}-15 \\ & \text { Hence\, common\, values \, of\, } \mathrm{a} \text { ' are }(2 \sqrt{5}-15,-\sqrt{5}-1) . \end{aligned} }
 

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HARSH KANKARIA

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