If the circle `x^2+y^2+4x+22y+l=0` bisects the circumference of the circle `x^2+y^2-2x+8y-m=0` , then l+m is equal to

Options

A. 60
B. 50
C. 40
D. 56

Answer

To solve the problem, we need to find the values of \( l \) and \( m \) such that the circle given by the equation \( x^2 + y^2 + 4x + 22y + l = 0 \) bisects the circumference of the circle given by the equation \( x^2 + y^2 - 2x + 8y - m = 0 \).

### Step 1: Identify the centers of the circles

The general form of a circle is given by the equation:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where the center of the circle is at the point \( (-g, -f) \).

For the first circle \( S_1: x^2 + y^2 + 4x + 22y + l = 0 \):
- Here, \( 2g = 4 \) and \( 2f = 22 \).
- Thus, the center of \( S_1 \) is \( (-2, -11) \).

For the second circle \( S_2: x^2 + y^2 - 2x + 8y - m = 0 \):
- Here, \( 2g = -2 \) and \( 2f = 8 \).
- Thus, the center of \( S_2 \) is \( (1, -4) \).

### Step 2: Find the equation of the line that bisects the circles

Since circle \( S_1 \) bisects the circumference of circle \( S_2 \), the line that bisects the circles must pass through the center of circle \( S_2 \) and be perpendicular to the line joining the centers of both circles.

The slope of the line joining the centers \( (-2, -11) \) and \( (1, -4) \) is:
\[ \text{slope} = \frac{-4 - (-11)}{1 - (-2)} = \frac{7}{3} \]
Thus, the slope of the perpendicular bisector is:
\[ \text{slope} = -\frac{3}{7} \]

### Step 3: Equation of the bisector line

Using the point-slope form of the equation of a line, the equation of the line passing through the center of \( S_2 \) (1, -4) with a slope of \( -\frac{3}{7} \) is:
\[ y + 4 = -\frac{3}{7}(x - 1) \]
Multiplying through by 7 to eliminate the fraction:
\[ 7y + 28 = -3x + 3 \implies 3x + 7y + 25 = 0 \]

### Step 4: Substitute the center of \( S_1 \) into the bisector equation

Now, we will substitute the center of \( S_1 \) \((-2, -11)\) into the bisector equation to find the relationship between \( l \) and \( m \):
\[ 3(-2) + 7(-11) + 25 = 0 \]
Calculating this gives:
\[ -6 - 77 + 25 = 0 \implies -58 + 25 = -33 \neq 0 \]
This means we need to express the relationship between \( l \) and \( m \) using the condition that the line passes through the center of \( S_2 \).

### Step 5: Set up the equation

The difference of the two circles \( S_1 - S_2 = 0 \) gives:
\[ (4 + 2) + (22 - 8) + (l + m) = 0 \]
Simplifying this:
\[ 6x + 14y + (l + m) = 0 \]
Substituting the center of \( S_2 \) into this equation:
\[ 6(1) + 14(-4) + (l + m) = 0 \]
Calculating gives:
\[ 6 - 56 + (l + m) = 0 \implies l + m = 50 \]

### Final Result

Thus, the value of \( l + m \) is:
\[ \boxed{50} \]