A circle passing through the intersection of the circles `x ^(2) + y^(2) + 5x + 4=0 and x ^(2) + y ^(2) + 5y -4 =0` also passes through the origin. The centre of the circle is

Options

A. `((5)/(2), (5)/(2))`
B. `((5)/(4), (5)/(4))`
C. `(- (5)/(4) , - (5)/(4))`
D. `(- (5)/(2) , -(5)/(2))`

Answer

To find the center of the circle that passes through the intersection of the given circles and also through the origin, we will follow these steps:

### Step 1: Write down the equations of the given circles.
The equations of the circles are:
1. \( S_1: x^2 + y^2 + 5x + 4 = 0 \)
2. \( S_2: x^2 + y^2 + 5y - 4 = 0 \)

### Step 2: Find the intersection of the two circles.
To find the equation of the circle that passes through the intersection of both circles, we can use the formula:
\[ S_1 + \lambda S_2 = 0 \]
where \( \lambda \) is a parameter that we will determine.

### Step 3: Substitute the equations into the formula.
Substituting \( S_1 \) and \( S_2 \) into the equation:
\[ (x^2 + y^2 + 5x + 4) + \lambda (x^2 + y^2 + 5y - 4) = 0 \]
This simplifies to:
\[ (1 + \lambda)x^2 + (1 + \lambda)y^2 + 5x + 5\lambda y + (4 - 4\lambda) = 0 \]

### Step 4: Since the circle passes through the origin, substitute \( (0, 0) \) into the equation.
Substituting \( (0, 0) \):
\[ (4 - 4\lambda) = 0 \]
This gives us:
\[ 4 - 4\lambda = 0 \implies \lambda = 1 \]

### Step 5: Substitute \( \lambda \) back into the equation.
Now substituting \( \lambda = 1 \) back into the equation:
\[ (1 + 1)x^2 + (1 + 1)y^2 + 5x + 5(1)y + (4 - 4(1)) = 0 \]
This simplifies to:
\[ 2x^2 + 2y^2 + 5x + 5y = 0 \]

### Step 6: Divide the entire equation by 2.
Dividing by 2 gives:
\[ x^2 + y^2 + \frac{5}{2}x + \frac{5}{2}y = 0 \]

### Step 7: Rearrange the equation to find the center.
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From our equation, we can identify:
- \( 2g = \frac{5}{2} \) → \( g = \frac{5}{4} \)
- \( 2f = \frac{5}{2} \) → \( f = \frac{5}{4} \)

### Step 8: Determine the center of the circle.
The center of the circle is given by the coordinates \( (-g, -f) \):
\[ \text{Center} = \left(-\frac{5}{4}, -\frac{5}{4}\right) \]

### Final Answer:
The center of the circle is \( \left(-\frac{5}{4}, -\frac{5}{4}\right) \).

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