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 \( R \) is equal to:

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

Hint:

The intersection of perpendicular diameters gives the center of the circle. The distance from the center to any point on the circle (including the origin, if it lies on the circle) is the radius.

Answer 1

The Correct Option is B

Step 1: Find the center of the circle by intersecting the diameters.
Solving \( x - y = 0 \) and \( x + y = 1 \) gives the center \( C \left( \frac{1}{2}, \frac{1}{2} \right) \).

Step 2: Use the condition that the circle passes through the origin \( (0, 0) \).
The radius \( R \) is the distance between the center and the origin.
Step 3: Calculate the distance.
\[ R = \sqrt{\left( \frac{1}{2} - 0 \right)^2 + \left( \frac{1}{2} - 0 \right)^2} = \sqrt{\left( \frac{1}{2} \right)^2 + \left( \frac{1}{2} \right)^2} = \sqrt{\frac{1}{4} + \frac{1}{4}} = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}} \]