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  • Three concentric circles of which biggest is , have their radii in A.P. If the line <img alt="y

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

Option: 1

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


Option: 2

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


Option: 3

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


Option: 4

\left(0, \frac{1}{3}-\frac{1}{3 \sqrt{2}}\right)


Answers (1)

best_answer

If ' d ' be the common difference of A.P. then radius of the smallest circle is \mathrm{1-2 d }. If the given line \mathrm{y-x-1=0 } cuts the smallest circle in real and distinct points then it will definitely cut the remaining circles in real and distinct points.

\mathrm{\begin{array}{ll} \Rightarrow & \quad \frac{|0-0-1|}{\sqrt{2}}<(1-2 \mathrm{~d}) \\ \Rightarrow & 1-2 \mathrm{~d}>\frac{1}{\sqrt{2}} \\ \Rightarrow & \mathrm{d}<\frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right) \\ \text { Hence } & \mathrm{d} \in\left(0, \frac{1}{2}\left(1-\frac{1}{\sqrt{2}}\right)\right) . \end{array} }

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Divya Prakash Singh

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