Question

Let \( x-y=0 \) and \( x+y=1 \) be two perpendicular diameters of a circle of radius \( R \). The circle will pass through the origin if \( R \) equals:

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{\sqrt{2}} \)
• C. \( \frac{1}{\sqrt{3}} \)
• D. \( \frac{1}{3} \)

Hint:

Perpendicular diameters: \begin{itemize} \item Their intersection is the center. \item Use distance formula to find radius. \end{itemize}

Answer 1

The Correct Option is B

Concept: Intersection of perpendicular diameters is center of circle. Step 1: {\color{red}Find center.} Solve: \[ x-y=0,\quad x+y=1 \] Add: \[ 2x=1 \Rightarrow x=\frac{1}{2}, \quad y=\frac{1}{2} \] Center = \( (\frac{1}{2},\frac{1}{2}) \). Step 2: {\color{red}Distance to origin.} Radius = distance from center to origin: \[ R = \sqrt{\frac{1}{4} + \frac{1}{4}} = \frac{1}{\sqrt{2}} \]