Question

The equation of the circle passing through the point $(1, 1)$ and having two diameters along the pair of lines $x^2 - y^2 - 2x + 4y - 3 = 0$ is

• A. $(x + 2)^2 + (y - 2)^2 = 4$
• B. $(x - 3)^2 + (y - 1)^2 = 4$
• C. $(x - 1)^2 + (y - 2)^2 = 1$
• D. $(x + 1)^2 + (y + 2)^2 = 1$

Hint:

To find the intersection of a pair of lines $f(x,y)=0$, solve $\partial f/\partial x = 0$ and $\partial f/\partial y = 0$.

Answer 1

The Correct Option is C

Step 1: Concept
The intersection of two diameters is the center of the circle.

Step 2: Meaning
Solve the joint equation $x^2 - y^2 - 2x + 4y - 3 = 0$ to find individual lines or find the point of intersection using partial derivatives.

Step 3: Analysis
$\frac{\partial f}{\partial x} = 2x - 2 = 0 \implies x = 1$. $\frac{\partial f}{\partial y} = -2y + 4 = 0 \implies y = 2$. Center is $(1, 2)$. Radius squared $r^2$ is the distance from $(1, 2)$ to $(1, 1)$: $r^2 = (1-1)^2 + (1-2)^2 = 1$.

Step 4: Conclusion
Equation is $(x - 1)^2 + (y - 2)^2 = 1$. Final Answer: (C)