The intercept made by the circle `x^(2)+y^(2)-4x-6y-3=0` on the line `x+y-3=0` is

Options

A. `sqrt(14)`
B. `2sqrt(14)`
C. `6sqrt(2)`
D. `3sqrt(2)`

Answer

To find the length of the intercept made by the circle \( x^2 + y^2 - 4x - 6y - 3 = 0 \) on the line \( x + y - 3 = 0 \), we can follow these steps:

### Step 1: Rewrite the line equation
The line equation can be rewritten to express \( y \) in terms of \( x \):
\[ y = 3 - x \]

### Step 2: Substitute \( y \) into the circle equation
Now, we substitute \( y = 3 - x \) into the circle equation:
\[ x^2 + (3 - x)^2 - 4x - 6(3 - x) - 3 = 0 \]

### Step 3: Expand and simplify the equation
Expanding \( (3 - x)^2 \):
\[ (3 - x)^2 = 9 - 6x + x^2 \]
Substituting this back into the equation gives:
\[ x^2 + (9 - 6x + x^2) - 4x - 18 + 6x - 3 = 0 \]
Combining like terms:
\[ 2x^2 - 4x + 9 - 18 - 3 = 0 \implies 2x^2 - 4x - 12 = 0 \]

### Step 4: Divide the equation by 2
To simplify, divide the entire equation by 2:
\[ x^2 - 2x - 6 = 0 \]

### Step 5: Solve for \( x \) using the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -2, c = -6 \):
\[ x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-6)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{2 \pm \sqrt{4 + 24}}{2} = \frac{2 \pm \sqrt{28}}{2} = \frac{2 \pm 2\sqrt{7}}{2} = 1 \pm \sqrt{7} \]

### Step 6: Find the corresponding \( y \) values
Now, substitute \( x \) back into the line equation to find \( y \):
For \( x = 1 + \sqrt{7} \):
\[ y = 3 - (1 + \sqrt{7}) = 2 - \sqrt{7} \]
For \( x = 1 - \sqrt{7} \):
\[ y = 3 - (1 - \sqrt{7}) = 2 + \sqrt{7} \]

### Step 7: Identify the intercept points
The intercept points on the line are:
1. \( (1 + \sqrt{7}, 2 - \sqrt{7}) \)
2. \( (1 - \sqrt{7}, 2 + \sqrt{7}) \)

### Step 8: Calculate the length of the intercept
The length of the intercept can be calculated using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the points:
\[ \text{Length} = \sqrt{((1 + \sqrt{7}) - (1 - \sqrt{7}))^2 + ((2 - \sqrt{7}) - (2 + \sqrt{7}))^2} \]
This simplifies to:
\[ = \sqrt{(2\sqrt{7})^2 + (-2\sqrt{7})^2} = \sqrt{4 \cdot 7 + 4 \cdot 7} = \sqrt{56 + 56} = \sqrt{112} = 4\sqrt{7} \]

### Final Answer
The length of the intercept made by the circle on the line is:
\[ \text{Length of the intercept} = 4\sqrt{7} \]