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Three concentric circles of which the biggest is \mathrm{x^{2}+y^{2}=1}, have their radii in A.P. If the line \mathrm{y=x+1} cuts all the circles in real and distinct points. The interval in which the common difference of the A.P. will lie is

Option: 1

\left(0, \frac{1}{4}\right)


Option: 2

\left(0, \frac{1}{2 \sqrt{2}}\right)


Option: 3

\left(0, \frac{2-\sqrt{2}}{4}\right)


Option: 4

none


Answers (1)

best_answer

Radius of circle are \mathrm{r_{1}, r_{2}} and 1
\mathrm{line \, y=x+1}

perpendicular from \mathrm{(0,0)} on line \mathrm{y=x+1}
\mathrm{=\frac{1}{\sqrt{2}}}
\mathrm{now \quad r_{1}>\frac{1}{\sqrt{2}} \Rightarrow r_{1}=1-2 d \Rightarrow \frac{1-r_{1}}{2}=d}
\mathrm{\therefore \quad \mathrm{d}=\frac{\sqrt{2}-1}{2 \sqrt{2}}}

Aliter : Equation of circle are
          
 \mathrm{x^{2}+y^{2}=1}
\mathrm{x^{2}+y^{2}=(1-d)^{2}}
\mathrm{x^{2}+y^{2}=(1-2 d)^{2}}

\mathrm{\Rightarrow \quad}  solve any of circle with line \mathrm{y=x+1}
\mathrm{e.g. x^{2}+y^{2}=(1-d)^{2} \Rightarrow 2 x^{2}+2 x+2 d-d^{2}=0} cuts the circle in real and distinct point hence \mathrm{\Delta>0}

\mathrm{\Rightarrow \quad 2 d^{2}-4 d+1>0 \quad \Rightarrow \quad d=\frac{2 \pm \sqrt{2}}{4} \quad}

Posted by

Pankaj Sanodiya

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