Question

The equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius is

• A. $x^{2}-y^{2}-x-y=0$
• B. $x^{2}+y^{2}-x-y=0$
• C. $x^{2}+y^{2}+x+y=0$
• D. $x^{2}+y^{2}-2x-2y=0$

Hint:

The equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius is

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For a circle passing through two points to have the minimum radius, the line segment joining the two points must be the diameter of the circle.
Step 2: Evaluation
Using the diameter form of a circle equation $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$ for points (1, 0) and (0, 1).
Step 3: Final Calculation
$(x-1)(x-0) + (y-0)(y-1) = 0 \Rightarrow x^2 - x + y^2 - y = 0$.
Step 4: Conclusion
The equation is $x^2 + y^2 - x - y = 0$.
Hence, the Answer is: (b)