Question

In the family of concentric circles $2(x^2 + y^2) = k$, the radius of the circle passing through $(1, 1)$ is:

• A. $\sqrt{2}$
• B. $4$
• C. $2\sqrt{2}$
• D. $1$
• E. $3\sqrt{2}$

Hint:

For circles centered at origin, radius = distance from origin to any point on circle: \[ R = \sqrt{x^2 + y^2} \]

Answer 1

The Correct Option is A

Concept: All circles of the form \( 2(x^2 + y^2) = k \) are concentric with center at origin. To find the radius, first determine \( k \), then convert to standard form: \[ x^2 + y^2 = R^2 \]

Step 1: Find value of \( k \).
Given point \( (1,1) \) lies on the circle: \[ 2(1^2 + 1^2) = k \] \[ 2(2) = k \Rightarrow k = 4 \]

Step 2: Write equation of the circle.
\[ 2(x^2 + y^2) = 4 \] Divide by 2: \[ x^2 + y^2 = 2 \]

Step 3: Find the radius.
Comparing with standard form: \[ x^2 + y^2 = R^2 \] \[ R^2 = 2 \Rightarrow R = \sqrt{2} \]

Step 4: Final answer.
\[ \boxed{\sqrt{2}} \]