Find the length of the chord intercepted by the circle `x^(2)+y^(2)-8x-6y=0` on the line `x-7y-8=0`.

Options

A. 5
B. `sqrt(2)`
C. `5sqrt(2)`
D. 6

Answer

To find the length of the chord intercepted by the circle \( x^2 + y^2 - 8x - 6y = 0 \) on the line \( x - 7y - 8 = 0 \), we will follow these steps:

### Step 1: Rewrite the Circle Equation
We start with the equation of the circle:
\[ x^2 + y^2 - 8x - 6y = 0 \]
We will complete the square for both \( x \) and \( y \).

1. For \( x^2 - 8x \), we add and subtract \( 16 \) (which is \( (8/2)^2 \)):
\[ x^2 - 8x + 16 - 16 \]
This becomes \( (x - 4)^2 - 16 \).

2. For \( y^2 - 6y \), we add and subtract \( 9 \) (which is \( (6/2)^2 \)):
\[ y^2 - 6y + 9 - 9 \]
This becomes \( (y - 3)^2 - 9 \).

Putting it all together, we have:
\[ (x - 4)^2 + (y - 3)^2 - 25 = 0 \]
Thus, the equation of the circle is:
\[ (x - 4)^2 + (y - 3)^2 = 25 \]
This shows that the center of the circle is \( (4, 3) \) and the radius is \( 5 \).

### Step 2: Find the Perpendicular Distance from the Center to the Line
Next, we need to find the perpendicular distance from the center of the circle \( (4, 3) \) to the line \( x - 7y - 8 = 0 \).

The formula for the distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1 \), \( B = -7 \), \( C = -8 \), and the point is \( (4, 3) \).

Substituting these values:
\[ d = \frac{|1 \cdot 4 - 7 \cdot 3 - 8|}{\sqrt{1^2 + (-7)^2}} = \frac{|4 - 21 - 8|}{\sqrt{1 + 49}} = \frac{|-25|}{\sqrt{50}} = \frac{25}{5\sqrt{2}} = \frac{5}{\sqrt{2}} \]

### Step 3: Use Pythagorean Theorem to Find Half the Chord Length
Let \( O \) be the center of the circle, \( L \) be the foot of the perpendicular from \( O \) to the line, and \( A \) and \( B \) be the endpoints of the chord. By the Pythagorean theorem in triangle \( OAL \):
\[ OA^2 = OL^2 + AL^2 \]
Where:
- \( OA = 5 \) (radius)
- \( OL = \frac{5}{\sqrt{2}} \) (perpendicular distance)

Thus,
\[ 5^2 = \left(\frac{5}{\sqrt{2}}\right)^2 + AL^2 \]
Calculating:
\[ 25 = \frac{25}{2} + AL^2 \]
\[ AL^2 = 25 - \frac{25}{2} = \frac{50}{2} - \frac{25}{2} = \frac{25}{2} \]
\[ AL = \sqrt{\frac{25}{2}} = \frac{5}{\sqrt{2}} \]

### Step 4: Find the Length of the Chord
The length of the chord \( AB \) is twice the length of \( AL \):
\[ AB = 2 \cdot AL = 2 \cdot \frac{5}{\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2} \]

### Final Answer
Thus, the length of the chord intercepted by the circle on the given line is:
\[ \boxed{5\sqrt{2}} \]