Question

The equation of the circle concentric with circle \(x^2+y^2-6x+7 = 0\) and which touches the line \(x+y+3 = 0\) is ....

• A. \(x^2+y^2-6x+9 = 0\)
• B. \(x^2+y^2-6x-9 = 0\)
• C. \(x^2+y^2-6x+3 = 0\)
• D. \(x^2+y^2-6x-3 = 0\)

Hint:

Same centre \((3,0)\); radius equals the distance from the centre to the tangent line.

Answer 1

The Correct Option is B

Step 1: Plan:
Write the new circle as $x^2+y^2-6x+c = 0$, and use the condition that the line is a tangent.

Step 2: Condition:
The centre is $(3,0)$ and $r^2 = 9 - c$. Distance from the centre to the line is $\frac{6}{\sqrt2}$, so $r^2 = \frac{36}{2} = 18$.
Therefore $9 - c = 18$, and $c = -9$.

Final Answer:
The circle is $x^2+y^2-6x-9=0$, option (B). \[ \boxed{x^2+y^2-6x-9=0} \]