The length of the chord x + y = 3 intercepted by the circle `x^2 + y^2 - 2x- 2y - 2 =0` is

Options

A. `7/2`
B. `3sqrt3/2`
C. `sqrt(14)`
D. `(sqrt7)/(2)`

Answer

To find the length of the chord \( x + y = 3 \) intercepted by the circle given by the equation \( x^2 + y^2 - 2x - 2y - 2 = 0 \), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we will rewrite the equation of the circle in standard form. The given equation is:
\[ x^2 + y^2 - 2x - 2y - 2 = 0 \]
We can rearrange it as:
\[ x^2 - 2x + y^2 - 2y = 2 \]
Now, we complete the square for \( x \) and \( y \).

### Step 2: Completing the Square
For \( x^2 - 2x \):
\[ x^2 - 2x = (x - 1)^2 - 1 \]
For \( y^2 - 2y \):
\[ y^2 - 2y = (y - 1)^2 - 1 \]
Substituting these back into the equation gives:
\[ (x - 1)^2 - 1 + (y - 1)^2 - 1 = 2 \]
Simplifying this:
\[ (x - 1)^2 + (y - 1)^2 - 2 = 2 \]
\[ (x - 1)^2 + (y - 1)^2 = 4 \]
This shows that the circle has a center at \( (1, 1) \) and a radius of \( 2 \).

### Step 3: Finding the Perpendicular Distance from the Center to the Chord
The line \( x + y = 3 \) can be rewritten in the form \( Ax + By + C = 0 \):
\[ x + y - 3 = 0 \quad \Rightarrow \quad A = 1, B = 1, C = -3 \]
The center of the circle is \( (1, 1) \). We can find the perpendicular distance \( d \) from the center to the line using the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_1, y_1) = (1, 1) \):
\[ d = \frac{|1 \cdot 1 + 1 \cdot 1 - 3|}{\sqrt{1^2 + 1^2}} = \frac{|1 + 1 - 3|}{\sqrt{2}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

### Step 4: Using Pythagoras' Theorem to Find Half the Chord Length
Let \( r \) be the radius of the circle, which is \( 2 \), and \( d \) be the distance from the center to the chord, which we found to be \( \frac{1}{\sqrt{2}} \). The relationship between the radius, the distance to the chord, and half the chord length \( L \) is given by:
\[ r^2 = d^2 + L^2 \]
Substituting the known values:
\[ 2^2 = \left(\frac{1}{\sqrt{2}}\right)^2 + L^2 \]
\[ 4 = \frac{1}{2} + L^2 \]
\[ L^2 = 4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2} \]
Thus,
\[ L = \sqrt{\frac{7}{2}} = \frac{\sqrt{7}}{\sqrt{2}} = \frac{\sqrt{14}}{2} \]

### Step 5: Finding the Total Length of the Chord
The total length of the chord is \( 2L \):
\[ \text{Length of the chord} = 2L = 2 \cdot \frac{\sqrt{14}}{2} = \sqrt{14} \]

### Final Answer
The length of the chord \( x + y = 3 \) intercepted by the circle is \( \sqrt{14} \).