Geometry: Chord Length Calculation

We need to find the length of the chord formed by the intersection of the line $y = x - 1$ and the circle with center $(1, 1)$ and radius $r = 1$. We can use the relationship between the radius, the distance from the circle's center to the chord, and half the chord length.

Distance from Center to Chord

First, rewrite the line equation in the general form $Ax + By + C = 0$: $x - y - 1 = 0$. The center of the circle is $(x_0, y_0) = (1, 1)$. The radius is $r = 1$. The distance ($d$) from the center $(1, 1)$ to the line $x - y - 1 = 0$ is calculated using the formula:

$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$ $d = \frac{|1(1) + (-1)(1) + (-1)|}{\sqrt{1^2 + (-1)^2}}$ $d = \frac{|1 - 1 - 1|}{\sqrt{1 + 1}}$ $d = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

Chord Length Calculation

Let the length of the chord be $L$. The radius ($r$), the distance ($d$), and half the chord length ($L/2$) form a right-angled triangle, with the radius as the hypotenuse.

Using the Pythagorean theorem:

$r^2 = d^2 + (L/2)^2$ $1^2 = \left(\frac{1}{\sqrt{2}}\right)^2 + (L/2)^2$ $1 = \frac{1}{2} + (L/2)^2$

Now, solve for $(L/2)^2$:

$(L/2)^2 = 1 - \frac{1}{2} = \frac{1}{2}$

Find $L/2$:

$L/2 = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}

Finally, calculate the full chord length $L$:

$L = 2 \times (L/2) = 2 \times \frac{1}{\sqrt{2}} = \sqrt{2}$

Final Result

The length of the chord is $\sqrt{2}$. Rounded to three decimal places, this is $1.414$.