Question

The radical axis of two orthogonal circles is \(x+1=0\). If one of those circles is \(x^2+y^2=4\), then the equation of the other circle is:

• A. \(x^2+y^2+8x+4=0\)
• B. \(x^2+y^2-4x+4=0\)
• C. \(x^2+y^2-8x+2y+8=0\)
• D. \(x^2+y^2-16=0\)

Hint:

To find the radical axis of two circles, subtract their equations directly. For orthogonal circles always use: \[ 2g_1g_2+2f_1f_2=c_1+c_2 \]

Answer 1

The Correct Option is A

Step 1: Recall two key circle ideas.
Two circles are orthogonal (cut at right angles) when $2g_1g_2+2f_1f_2=c_1+c_2$. The radical axis is found by subtracting the two circle equations.
Step 2: Write the known circle in full form.
$x^2+y^2=4$ is $x^2+y^2-4=0$, so $g_1=0$, $f_1=0$, $c_1=-4$.
Step 3: Set up the unknown circle.
Let it be $x^2+y^2+2gx+2fy+c=0$. Subtracting the known circle gives the radical axis: \[ 2gx+2fy+c+4=0. \]
Step 4: Match it to the given radical axis.
The radical axis is $x+1=0$. There is no $y$, so $2f=0$, giving $f=0$.
Step 5: Use orthogonality to find $c$.
With $g_1=f_1=0$, the orthogonal rule becomes $0=c_1+c=-4+c$, so $c=4$.
Step 6: Find $g$ and write the circle.
Now the radical axis is $2gx+8=0$. Matching $x+1=0$ means $2gx+8$ is a multiple of $x+1$: the ratio gives $2g=8$, so $g=4$. The circle is \[ \boxed{x^2+y^2+8x+4=0}. \]