Question:
The image of the point \((3,4)\) with respect to the radical axis of the circles
\[
x^2+y^2+8x+2y+10=0
\]
and
\[
x^2+y^2+7x+3y+10=0
\]
is
The image of the point \((3,4)\) with respect to the radical axis of the circles
\[
x^2+y^2+8x+2y+10=0
\]
and
\[
x^2+y^2+7x+3y+10=0
\]
is
Show Hint
The radical axis of two circles is found by subtracting their equations. Also, reflection of \((a,b)\) about the line \(x=y\) is \((b,a)\).
Updated On: Jun 22, 2026
- \((3,4)\)
- \((-4,-3)\)
- \((4,3)\)
- \((-3,-4)\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Find the radical axis of the two circles.
The given circles are
\[ S_1:x^2+y^2+8x+2y+10=0 \] and
\[ S_2:x^2+y^2+7x+3y+10=0 \] The radical axis is obtained by subtracting \(S_2\) from \(S_1\):
\[ S_1-S_2=0 \] \[ (x^2+y^2+8x+2y+10)-(x^2+y^2+7x+3y+10)=0 \] \[ x-y=0 \] So, the radical axis is
\[ x-y=0 \] or
\[ x=y \]
Step 2: Understand the reflection about the line \(x=y\).
The image of any point \((a,b)\) with respect to the line \(x=y\) is obtained by interchanging its coordinates.
So,
\[ (a,b)\rightarrow (b,a) \]
Step 3: Find the image of \((3,4)\).
Given point is
\[ (3,4) \] Reflecting it about the line \(x=y\), we interchange the coordinates:
\[ (3,4)\rightarrow (4,3) \]
Step 4: Final conclusion.
Hence, the image of the point \((3,4)\) with respect to the radical axis is
\[ \boxed{(4,3)} \]
The given circles are
\[ S_1:x^2+y^2+8x+2y+10=0 \] and
\[ S_2:x^2+y^2+7x+3y+10=0 \] The radical axis is obtained by subtracting \(S_2\) from \(S_1\):
\[ S_1-S_2=0 \] \[ (x^2+y^2+8x+2y+10)-(x^2+y^2+7x+3y+10)=0 \] \[ x-y=0 \] So, the radical axis is
\[ x-y=0 \] or
\[ x=y \]
Step 2: Understand the reflection about the line \(x=y\).
The image of any point \((a,b)\) with respect to the line \(x=y\) is obtained by interchanging its coordinates.
So,
\[ (a,b)\rightarrow (b,a) \]
Step 3: Find the image of \((3,4)\).
Given point is
\[ (3,4) \] Reflecting it about the line \(x=y\), we interchange the coordinates:
\[ (3,4)\rightarrow (4,3) \]
Step 4: Final conclusion.
Hence, the image of the point \((3,4)\) with respect to the radical axis is
\[ \boxed{(4,3)} \]
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