Introduction

Understanding the relationship between two intersecting circles is essential in various mathematical applications, including geometry and algebra. In this article, we will explore the process of finding the equation of the common chord and the intersection points of two given circles. This topic is crucial for students and professionals in mathematics and related fields.

The Problem

We are given the following equations of the circles:

Circle 1: x2 y2-4x-2y 10 Circle 2: x2 y2-4x-6y-100

The objective is to find the equation of the common chord and the points of intersection between these circles.

Step 1: Simplify the Equations of the Circles

First, let's complete the square for each equation to identify the centers and radii of the circles:

Circle 1

Rewrite the equation of Circle 1 as:

(x^2 - 4x y^2 - 2y 1 0)

Complete the square for (x) and (y):

[begin{align*} (x - 2)^2 - 4 (y - 1)^2 - 1 1 0 (x - 2)^2 (y - 1)^2 4 end{align*}]

Therefore, Circle 1 has a center at ((2, 1)) and a radius of (2).

Circle 2

Rewrite the equation of Circle 2 as:

(x^2 - 4x y^2 - 6y - 10 0)

Complete the square for (x) and (y):

[begin{align*} (x - 2)^2 - 4 (y - 3)^2 - 9 - 10 0 (x - 2)^2 (y - 3)^2 23 end{align*}]

Therefore, Circle 2 has a center at ((-2, 3)) and a radius of (sqrt{23}).

Step 2: Find the Equation of the Common Chord

To find the equation of the common chord, we subtract the equation of Circle 2 from the equation of Circle 1:

[begin{align*} (x^2 - 4x - 2y 10) - (x^2 - 4x - 6y - 10) 0 8y 20 0 4y -10 y -frac{5}{2} or 8x - 4y - 11 0 end{align*}]

The equation of the common chord is (8x - 4y - 11 0).

Step 3: Solve for Intersection Points

Solve for the intersection points by substituting the common chord equation into one of the circle equations. Let's substitute into Circle 1:

[begin{align*} 8x - 4left(-frac{5}{2}right) - 11 0 8x 10 - 11 0 8x - 1 0 x frac{1}{8} end{align*}]

Now, substitute (x frac{1}{8}) into the common chord equation to find (y):

[begin{align*} 8left(frac{1}{8}right) - 4y - 11 0 1 - 4y - 11 0 -4y - 10 0 y -frac{5}{2} end{align*}]

This yields one point of intersection. Next, let's solve the common chord equation for (x) and (y) to find the other point of intersection:

Solve (8x - 4y - 11 0) for (y):

[y frac{8x - 11}{4}]

Substitute into Circle 1:

[begin{align*}(x - 2)^2 left(frac{8x - 11}{4} - 1right)^2 4 (x - 2)^2 left(frac{8x - 15}{4}right)^2 4 (x - 2)^2 frac{(8x - 15)^2}{16} 4 16(x - 2)^2 (8x - 15)^2 64 16(x^2 - 4x 4) (64x^2 - 24 225) 64 16x^2 - 64x 64 64x^2 - 24 225 64 8^2 - 304x 225 0 2^2 - 76x 56.25 0 (2x - 1)(1 - 27) 0 end{align*}]

Solving the quadratic equation, we get:

[x frac{1}{2} quad or quad x frac{27}{10}]

Substitute these (x) values back into the common chord equation to find the corresponding (y) values:

For (x frac{1}{2}): [y 8left(frac{1}{2}right) - 11 4 - 11 -7] For (x frac{27}{10}): [y 8left(frac{27}{10}right) - 11 frac{216}{10} - 11 21.6 - 11 10.6]

Thus, the points of intersection are (left(frac{1}{2}, -7right)) and (left(frac{27}{10}, 10.6right)).

Conclusion

In summary, the equation of the common chord of the circles (x^2 y^2 - 4x - 2y 1 0) and (x^2 y^2 - 4x - 6y - 10 0) is (8x - 4y - 11 0). The points of intersection are (left(frac{1}{8}, -frac{5}{2}right)) and (left(frac{27}{10}, 10.6right)).