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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}, \mathrm{r}_{2} and 1

\mathrm{line \: y=x+1}

\text{perpendicular from }\mathrm{(0,0) on\: line\: \mathrm{y}=\mathrm{x}+1}

\mathrm{=\frac{1}{\sqrt{2}}}
\mathrm{now \: \mathrm{r}_{1}>\frac{1}{\sqrt{2}} \Rightarrow \mathrm{r}_{1}=1-2 \mathrm{~d} \Rightarrow \frac{1-\mathrm{r}_{1}}{2}=\mathrm{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 \text{solve any of circle with line}\, \mathrm{y}=\mathrm{x}+1}

\mathrm{e.g. \quad x^{2}+y^{2}=(1-d)^{2} \Rightarrow 2 x^{2}+2 x+2 d-d^{2}=0\text{ cuts are real and distinct point hence}\, \Delta>0}
\mathrm{\Rightarrow \quad 2 \mathrm{~d}^{2}-4 \mathrm{~d}+1>0 \quad \Rightarrow \quad \mathrm{d}=\frac{2 \pm \sqrt{2}}{4} \quad}




 

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vishal kumar

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