Question

The equation of a circle whose Centre is (2, -1) and which passes through the point (3, 6) is

• A. $x^{2} + y^{2} + 4x + 2y - 45 = 0$
• B. $x^{2} + y^{2} - 2x + 2y - 50 = 0$
• C. $x^{2} + y^{2} + 2x + 2y - 50 = 0$
• D. $x^{2} + y^{2} - 4x + 2y - 45 = 0$

Hint:

If the center is $(h, k)$, the linear terms in the general form are always $-2hx$ and $-2ky$. For $(2, -1)$, look for $-4x$ and $+2y$.

Answer 1

The Correct Option is D

Step 1: Concept
The radius $r$ is the distance between the center $(h, k)$ and any point $(x_1, y_1)$ on the circle.

Step 2: Meaning
Calculate $r^2$ using the distance formula: $r^{2} = (x_1 - h)^{2} + (y_1 - k)^{2}$.

Step 3: Analysis
$r^{2} = (3 - 2)^{2} + (6 - (-1))^{2} = 1^{2} + 7^{2} = 1 + 49 = 50$. The circle equation is $(x - 2)^{2} + (y + 1)^{2} = 50$.

Step 4: Conclusion
Expanding gives $x^{2} - 4x + 4 + y^{2} + 2y + 1 = 50$, which simplifies to $x^{2} + y^{2} - 4x + 2y - 45 = 0$.
Final Answer: (D)