find the radius. pls qive the answer as one of this optios coz the ques is not incomplete or wrong this is kvpy ques

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Let} \mathrm{circle} \mathrm{cut} \mathrm{side} \mathrm{AB} \mathrm{at} \mathrm{E} \mathrm{and} \mathrm{touches} \mathrm{AB} \mathrm{at} \mathrm{F}. \\ \Rightarrow \mathrm{AF}=\frac{1}{2} \left(\mathrm{By} \mathrm{symmetry}\right) \\ \mathrm{Now} \mathrm{circle} \mathrm{passes} \mathrm{through} \mathrm{B} \mathrm{and} \mathrm{C} \mathrm{and} \\ \angle \mathrm{EBC}=90° \mathrm{which} \mathrm{means} \mathrm{that} \\ \mathrm{EC}\hspace{0.17em}\mathrm{is} \mathrm{diameter} \mathrm{of} \mathrm{circle} \left(\mathrm{Angle} \mathrm{in} \mathrm{semi} \mathrm{circle} \mathrm{is} 90°\right) \\ \mathrm{Now} \\ \mathrm{By} \mathrm{property} \mathrm{of} \mathrm{cirlce} \\ \mathrm{AE}\times \mathrm{AB}={\mathrm{AF}}^2 \\ \Rightarrow \mathrm{AE}\times 1=\frac{1}{4} \\ \Rightarrow \mathrm{AE}=\frac{1}{4} \\ \Rightarrow \mathrm{BE}=1-\frac{1}{4}=\frac{3}{4} \\ \mathrm{In} \mathrm{right} \mathrm{angled} \mathrm{triangle} \mathrm{EBC} \\ \mathrm{EC}=\sqrt{{\mathrm{EB}}^2+{\mathrm{BC}}^2}=\sqrt{\frac{9}{16}+1}=\sqrt{\frac{25}{16}}=\frac{5}{4} \\ \mathrm{Radius}=\frac{\mathrm{EC}}{2}=\frac{\frac{5}{4}}{2}=\frac{5}{8} \end{gathered}$$
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