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:
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:
Show Hint
To find the radical axis of two circles, subtract their equations directly.
For orthogonal circles always use:
\[
2g_1g_2+2f_1f_2=c_1+c_2
\]
Updated On: Jun 17, 2026
- \(x^2+y^2+8x+4=0\)
- \(x^2+y^2-4x+4=0\)
- \(x^2+y^2-8x+2y+8=0\)
- \(x^2+y^2-16=0\)
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The Correct Option is A
Solution and Explanation
Concept:
Two circles are said to be orthogonal if they intersect at right angles. If the equations of two circles are:
\[
x^2+y^2+2g_1x+2f_1y+c_1=0
\]
and
\[
x^2+y^2+2g_2x+2f_2y+c_2=0
\]
then the condition for orthogonality is:
\[
2g_1g_2+2f_1f_2=c_1+c_2
\]
The radical axis of two circles is obtained by subtracting their equations.
Step 1: Write the given circle in general form.
Given circle: \[ x^2+y^2=4 \] can be written as: \[ x^2+y^2-4=0 \] Thus, \[ g_1=0,\quad f_1=0,\quad c_1=-4 \]
Step 2: Use the radical axis condition.
The radical axis is: \[ x+1=0 \] Suppose the second circle is: \[ x^2+y^2+2gx+2fy+c=0 \] Subtracting the first circle from the second: \[ 2gx+2fy+c+4=0 \] This must represent the same line as \(x+1=0\). Hence, \[ 2f=0 \Rightarrow f=0 \] and \[ 2gx+c+4=0 \] Comparing with \(x+1=0\), \[ 2g=1k,\quad c+4=k \] for some constant \(k\).
Step 3: Apply orthogonality condition.
Since \(g_1=f_1=0\), \[ 0=c_1+c \] \[ 0=-4+c \] \[ c=4 \] Now, \[ c+4=8=k \] Thus, \[ 2g=8 \Rightarrow g=4 \] Therefore, the second circle is: \[ x^2+y^2+8x+4=0 \] Hence the correct answer is: \[ \boxed{x^2+y^2+8x+4=0} \]
Step 1: Write the given circle in general form.
Given circle: \[ x^2+y^2=4 \] can be written as: \[ x^2+y^2-4=0 \] Thus, \[ g_1=0,\quad f_1=0,\quad c_1=-4 \]
Step 2: Use the radical axis condition.
The radical axis is: \[ x+1=0 \] Suppose the second circle is: \[ x^2+y^2+2gx+2fy+c=0 \] Subtracting the first circle from the second: \[ 2gx+2fy+c+4=0 \] This must represent the same line as \(x+1=0\). Hence, \[ 2f=0 \Rightarrow f=0 \] and \[ 2gx+c+4=0 \] Comparing with \(x+1=0\), \[ 2g=1k,\quad c+4=k \] for some constant \(k\).
Step 3: Apply orthogonality condition.
Since \(g_1=f_1=0\), \[ 0=c_1+c \] \[ 0=-4+c \] \[ c=4 \] Now, \[ c+4=8=k \] Thus, \[ 2g=8 \Rightarrow g=4 \] Therefore, the second circle is: \[ x^2+y^2+8x+4=0 \] Hence the correct answer is: \[ \boxed{x^2+y^2+8x+4=0} \]
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