The equation of diameter of a circle `x^(2) + y^(2) + 2x - 4y =4`, that is parallel to `3x + 5y =4` is

Options

A. `3x+5y=-7`
B. `3x+5y=7`
C. `3x+5y=9`
D. `3x+5y=1 `

Answer

To solve the problem of finding the equation of the diameter of the circle given by the equation \(x^2 + y^2 + 2x - 4y = 4\) that is parallel to the line \(3x + 5y = 4\), we can follow these steps:

### Step 1: Rewrite the Circle's Equation
The given equation of the circle is:
\[ x^2 + y^2 + 2x - 4y = 4 \]
We can rewrite this in standard form by moving the constant to the other side:
\[ x^2 + y^2 + 2x - 4y - 4 = 0 \]

### Step 2: Identify the Center of the Circle
The general form of a circle is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From our equation, we can identify:
- \(2g = 2\) so \(g = 1\)
- \(2f = -4\) so \(f = -2\)

The center of the circle \((h, k)\) is given by \((-g, -f)\):
\[ \text{Center} = (-1, 2) \]

### Step 3: Find the Slope of the Given Line
The equation of the line is:
\[ 3x + 5y = 4 \]
We can rewrite this in slope-intercept form \(y = mx + b\):
\[ 5y = -3x + 4 \implies y = -\frac{3}{5}x + \frac{4}{5} \]
The slope \(m\) of this line is \(-\frac{3}{5}\).

### Step 4: Find the Slope of the Diameter
Since the diameter is parallel to the given line, it will have the same slope:
\[ \text{Slope of diameter} = -\frac{3}{5} \]

### Step 5: Write the Equation of the Diameter
Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the center of the circle \((-1, 2)\) and \(m = -\frac{3}{5}\):
\[ y - 2 = -\frac{3}{5}(x + 1) \]

### Step 6: Simplify the Equation
Now, we simplify this equation:
\[ y - 2 = -\frac{3}{5}x - \frac{3}{5} \]
Adding \(2\) to both sides:
\[ y = -\frac{3}{5}x - \frac{3}{5} + 2 \]
Convert \(2\) to a fraction:
\[ 2 = \frac{10}{5} \]
So:
\[ y = -\frac{3}{5}x + \frac{10}{5} - \frac{3}{5} = -\frac{3}{5}x + \frac{7}{5} \]

### Step 7: Write in Standard Form
To write it in standard form:
\[ 3x + 5y = 7 \]

Thus, the equation of the diameter of the circle that is parallel to the line \(3x + 5y = 4\) is:
\[ \boxed{3x + 5y = 7} \]