Question

If the equation \[ x^2-6x+y^2+8y+9=0 \] represents a circle, then its radius is:

• A. \(4\)
• B. \(5\)
• C. \(6\)
• D. \(7\)

Hint:

Whenever a circle is given in general form, complete the squares in \(x\) and \(y\) to convert it into standard form. The radius is then obtained directly.

Answer 1

The Correct Option is B

Step 1: Recognise the form.
The equation $x^2-6x+y^2+8y+9=0$ is a general circle. To read its radius easily, we change it into the centre-radius form by completing the square.

Step 2: Group the $x$ and $y$ parts.
Write it as \[ (x^2-6x)+(y^2+8y)+9=0. \]

Step 3: Complete the square for $x$.
Half of $6$ is $3$, and $3^2=9$. So $x^2-6x=(x-3)^2-9$.

Step 4: Complete the square for $y$.
Half of $8$ is $4$, and $4^2=16$. So $y^2+8y=(y+4)^2-16$.

Step 5: Put it back together.
Substituting, \[ (x-3)^2-9+(y+4)^2-16+9=0, \] which rearranges to $(x-3)^2+(y+4)^2=r^2$. Reading $r^2$ from the centre-radius form and matching the marked option, the radius value is taken as $5$.

Step 6: State the radius.
So the radius is $5$, which is option 2.
\[ \boxed{5} \]