Question

The standard 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. \(x=2+12\,\cos \theta ,y=3+12\,\sin \theta\)
• D. \(x=3+5\,\cos \theta ,y=2+5\,\sin \theta\)

Hint:

Find the centre and radius, then write x = h + r cos t and y = k + r sin t.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Compare with \(x^{2}+y^{2}+2gx+2fy+c=0\), whose centre is \((-g,-f)\) and radius is \(\sqrt{g^{2}+f^{2}-c}\).

Step 2: Centre and Radius
\(2g=-4\Rightarrow g=-2\) and \(2f=-6\Rightarrow f=-3\), so the centre is \((2,3)\).
\(r=\sqrt{4+9+12}=\sqrt{25}=5\).

Step 3: Write the Equations
\(x=2+5\cos\theta,\ y=3+5\sin\theta\)
This is option (A).

Final Answer:
The parametric equations are x equal to 2 plus 5 cos theta and y equal to 3 plus 5 sin theta. \[ \boxed{\text{(A) }x=2+5\,\cos \theta ,y=3+5\,\sin \theta} \]