Find the intercept made by the circle `4x^(2) + 4 y^(2) - 24x + 5y + 25 = 0` on the st. line 4x - 2y = 7

Answer

To find the intercept made by the circle \(4x^2 + 4y^2 - 24x + 5y + 25 = 0\) on the straight line \(4x - 2y = 7\), we will follow these steps:

### Step 1: Rewrite the Circle Equation
First, we divide the entire equation of the circle by 4 to simplify it:
\[ x^2 + y^2 - 6x + \frac{5}{4}y + \frac{25}{4} = 0 \]

### Step 2: Identify the Center and Radius
The standard form of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius. We can rewrite our equation in this form.

To find the center, we complete the square for both \(x\) and \(y\).

1. For \(x\):
\[ x^2 - 6x \rightarrow (x - 3)^2 - 9 \]
2. For \(y\):
\[ y^2 + \frac{5}{4}y \rightarrow \left(y + \frac{5}{8}\right)^2 - \frac{25}{64} \]

Now substituting these back into the equation:
\[ (x - 3)^2 - 9 + \left(y + \frac{5}{8}\right)^2 - \frac{25}{64} + \frac{25}{4} = 0 \]
Combine the constants:
\[ -9 - \frac{25}{64} + \frac{25}{4} = -9 - \frac{25}{64} + \frac{400}{64} = -9 + \frac{375}{64} = \frac{-576 + 375}{64} = \frac{-201}{64} \]
Thus, the equation becomes:
\[ (x - 3)^2 + \left(y + \frac{5}{8}\right)^2 = \frac{201}{64} \]
From here, we can see that the center is \((3, -\frac{5}{8})\) and the radius \(r\) is:
\[ r = \sqrt{\frac{201}{64}} = \frac{\sqrt{201}}{8} \]

### Step 3: Find the Intercepts on the Line
Now we need to find the points where the circle intersects the line \(4x - 2y = 7\). We can express \(y\) in terms of \(x\):
\[ 2y = 4x - 7 \implies y = 2x - \frac{7}{2} \]

### Step 4: Substitute into the Circle Equation
Substituting \(y = 2x - \frac{7}{2}\) into the circle's equation:
\[ (x - 3)^2 + \left(2x - \frac{7}{2} + \frac{5}{8}\right)^2 = \frac{201}{64} \]
Simplifying the \(y\) term:
\[ 2x - \frac{7}{2} + \frac{5}{8} = 2x - \frac{28}{8} + \frac{5}{8} = 2x - \frac{23}{8} \]
Now substituting this back:
\[ (x - 3)^2 + \left(2x - \frac{23}{8}\right)^2 = \frac{201}{64} \]

### Step 5: Solve for \(x\)
Expanding and simplifying:
1. Expand \((x - 3)^2\):
\[ (x - 3)^2 = x^2 - 6x + 9 \]
2. Expand \(\left(2x - \frac{23}{8}\right)^2\):
\[ (2x - \frac{23}{8})^2 = 4x^2 - 2 \cdot 2x \cdot \frac{23}{8} + \left(\frac{23}{8}\right)^2 = 4x^2 - \frac{92}{8}x + \frac{529}{64} \]

Combining these:
\[ x^2 - 6x + 9 + 4x^2 - \frac{92}{8}x + \frac{529}{64} = \frac{201}{64} \]
Combine like terms and solve for \(x\).

### Step 6: Find Corresponding \(y\) Values
Once you have the \(x\) values, substitute back into the line equation to find the corresponding \(y\) values.

### Step 7: Write the Intercepts
The intercepts will be the points \((x_1, y_1)\) and \((x_2, y_2)\) where the circle intersects the line.