Question

The parametric equations of the circle $x^2 + y^2 - 4x - 6y - 12 = 0$ are:

• A. $x = 2 + 5 \cos \theta, y = 3 + 5 \sin \theta$
• B. $x = -2 + 5 \cos \theta, y = -3 + 5 \sin \theta$
• C. $x = 2 + 25 \cos \theta, y = 3 + 25 \sin \theta$
• D. $x = 5 + 2 \cos \theta, y = 5 + 3 \sin \theta$

Hint:

Center is $(-g, -f)$ and Radius is $\sqrt{g^2+f^2-c}$. For this circle, $g=-2, f=-3, c=-12$.

Answer 1

The Correct Option is A

Step 1: Concept
Convert the general form of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ to the standard form $(x-h)^2 + (y-k)^2 = r^2$.

Step 2: Meaning
Completing the square for $x$ and $y$:
$(x^2 - 4x + 4) + (y^2 - 6y + 9) = 12 + 4 + 9$
$(x-2)^2 + (y-3)^2 = 25$

Step 3: Analysis
The center is $(h, k) = (2, 3)$ and the radius is $r = \sqrt{25} = 5$.
The parametric equations are $x = h + r \cos \theta$ and $y = k + r \sin \theta$.

Step 4: Conclusion
Substituting the values: $x = 2 + 5 \cos \theta$ and $y = 3 + 5 \sin \theta$. Final Answer: (A)