Question

Length of the common chord of the circles \((x-1)^{2}+(y+1)^{2}=c^{2}\) and \((x+1)^{2}+(y-1)^{2}=c^{2}\) is (1) \(\frac{1}{2} \sqrt{c^{2}-2}\) (2) \(\sqrt{c^{2}-2}\) (3) \(c+2\) (4) \(2 \sqrt{c^{2}-2}\)

Short answer

The length is \( 2 \sqrt{ c^{2} - 2} \).

Step-by-step solution

- Identify the Center and Radius of Each Circle

The first circle \( (x-1)^2 + (y+1)^2 = c^2 \) has a center at \( (1, -1) \) and radius \( c \). The second circle \( (x+1)^2 + (y-1)^2 = c^2 \) has a center at \( (-1, 1) \) and radius \( c \).

- Calculate Distance Between Centers

The distance \( d \) between the centers of the circles is calculated using the distance formula: \[ d = \sqrt{(1 - (-1))^2 + (-1 - 1)^2} \] \[ d = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

- Find the Length of the Common Chord

The length \( L \) of the common chord of two intersecting circles can be found using the formula: \[ L = 2 \sqrt{ r^{2} - \left( \frac{d}{2} \right)^{2}} \] Here, radius \( r \) is \( c \), and distance \( d \) between centers is \( 2 \sqrt{2} \). Substituting these values: \[ L = 2 \sqrt{ c^{2} - \left( \frac{2 \sqrt{2}}{2} \right)^{2}} \] \[ L = 2 \sqrt{ c^{2} - (\sqrt{2})^{2}} \] \[ L = 2 \sqrt{ c^{2} - 2} \]

Key concepts

geometry

Geometry, especially circle geometry, is crucial in understanding the relationships between shapes and sizes. In our exercise, we are dealing with two circles. Each circle has a unique center and radius. When two circles intersect, the line segment that passes through both intersection points is called the common chord. This common chord is what we find here.

Understanding the geometric properties of circles, such as how to locate their centers and measure their radii, is the first step. Then, we use these properties to find other geometric relationships, like the distance between centers and the length of the common chord. Easy to understand visual aids like diagrams and images can be very helpful when studying such problems. Always try to draw the circles and axes to visualize the problem better.

circle equations

In our problem, we use the equations of two circles: \((x-1)^{2}+(y+1)^{2}=c^{2}\) and \((x+1)^{2}+(y-1)^{2}=c^{2}\). A circle’s equation in general form \( (x - h)^2 + (y - k)^2 = r^2 \) tells us about the circle's center and radius. Here:

• For \ (x-1)^{2}+(y+1)^{2}=c^{2} \: Center: \ (1, -1) \, Radius: \ c \
• For \ (x+1)^{2}+(y-1)^{2}=c^{2} \: Center: \ (-1, 1) \, Radius: \ c \

The circles' centers are the points where \ h \ and \ k \ variables appear in the equations. The radius is the square root of the right-hand side of the equation. Understanding circle equations helps us locate both the center and the radius accurately.

distance formula

To learn the distance between the centers of these circles, we use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Given the centers \ (1, -1) \ and \ (-1, 1) \, we get: \ d = \sqrt{(1 - (-1))^2 + (-1 - 1)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2 \sqrt{2} \ Using the coordinates, we account for the horizontal and vertical differences between the circle centers to calculate the distance. Applying the formula step by step ensures an accurate result.

chord length formula

Finally, to find the length of the common chord, we use this formula: \[ L = 2 \sqrt{r^2 - (\frac{d}{2})^2} \] Here, \ r \ is the circles' radius \ c \ and \ d \ is the distance between the centers \ 2\sqrt{2} \. Substituting, we get: \ L = 2 \sqrt{c^2 - (\frac{2\sqrt{2}}{2})^2} = 2 \sqrt{c^2 - (\sqrt{2})^2} = 2 \sqrt{c^2 - 2} \ This formula derives the common chord length based on the radius and the distance between the centers, illustrating a practical application of algebra with geometry. Every step is important to clarify how geometry and algebra interact.