Question:
If
\[
x^2+y^2-2x+2fy+c=0
\]
intersects the two circles
\[
x^2+y^2+2x-4y+1=0
\]
and
\[
x^2+y^2-4x-2y-11=0
\]
orthogonally, then
\[
f+c=
\]
If
\[
x^2+y^2-2x+2fy+c=0
\]
intersects the two circles
\[
x^2+y^2+2x-4y+1=0
\]
and
\[
x^2+y^2-4x-2y-11=0
\]
orthogonally, then
\[
f+c=
\]
Show Hint
For orthogonal circles,
\[
2g_1g_2+2f_1f_2=c_1+c_2.
\]
This relation is often much faster than converting the circles into centre-radius form.
Updated On: Jul 9, 2026
- \(20\)
- \(17\)
- \(12\)
- \(24\) \bigskip
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
Two circles
\[
x^2+y^2+2g_1x+2f_1y+c_1=0
\]
and
\[
x^2+y^2+2g_2x+2f_2y+c_2=0
\]
intersect orthogonally if
\[
2g_1g_2+2f_1f_2=c_1+c_2.
\]
Step 1: Apply the orthogonality condition with the first circle. The required circle is \[ x^2+y^2-2x+2fy+c=0. \] Hence, \[ g=-1, \qquad f=f. \] For \[ x^2+y^2+2x-4y+1=0, \] \[ g_1=1, \qquad f_1=-2, \qquad c_1=1. \] Using \[ 2gg_1+2ff_1=c+c_1, \] \[ 2(-1)(1)+2f(-2)=c+1. \] \[ -2-4f=c+1. \] \[ c=-3-4f. \] \[ \cdots (1) \]
Step 2: Apply the orthogonality condition with the second circle. For \[ x^2+y^2-4x-2y-11=0, \] \[ g_2=-2, \qquad f_2=-1, \qquad c_2=-11. \] Again, \[ 2gg_2+2ff_2=c+c_2. \] \[ 2(-1)(-2)+2f(-1)=c-11. \] \[ 4-2f=c-11. \] \[ c=15-2f. \] \[ \cdots (2) \]
Step 3: Solve for \(f\) and \(c\). From (1) and (2), \[ -3-4f=15-2f. \] \[ -18=2f. \] \[ f=-9. \] Substituting into (2), \[ c=15-2(-9). \] \[ c=33. \]
Step 4: Find \(f+c\). \[ f+c = -9+33. \] \[ =24. \]
Step 5: Write the final answer. \[ \boxed{24} \]
Step 1: Apply the orthogonality condition with the first circle. The required circle is \[ x^2+y^2-2x+2fy+c=0. \] Hence, \[ g=-1, \qquad f=f. \] For \[ x^2+y^2+2x-4y+1=0, \] \[ g_1=1, \qquad f_1=-2, \qquad c_1=1. \] Using \[ 2gg_1+2ff_1=c+c_1, \] \[ 2(-1)(1)+2f(-2)=c+1. \] \[ -2-4f=c+1. \] \[ c=-3-4f. \] \[ \cdots (1) \]
Step 2: Apply the orthogonality condition with the second circle. For \[ x^2+y^2-4x-2y-11=0, \] \[ g_2=-2, \qquad f_2=-1, \qquad c_2=-11. \] Again, \[ 2gg_2+2ff_2=c+c_2. \] \[ 2(-1)(-2)+2f(-1)=c-11. \] \[ 4-2f=c-11. \] \[ c=15-2f. \] \[ \cdots (2) \]
Step 3: Solve for \(f\) and \(c\). From (1) and (2), \[ -3-4f=15-2f. \] \[ -18=2f. \] \[ f=-9. \] Substituting into (2), \[ c=15-2(-9). \] \[ c=33. \]
Step 4: Find \(f+c\). \[ f+c = -9+33. \] \[ =24. \]
Step 5: Write the final answer. \[ \boxed{24} \]
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