Problem 21 - The Calculus with Analytic Geometry

Question

Find an equation of the common chord of the two circles \(x^{2}+y^{2}+4 x-6 y-12=0\) and \(x^{2}+y^{2}+8 x-2 y+8=0\). (HINT: If the coordinates of a point satisfy two different equations, then the coordinates also satisfy the difference of the two equations.)

Step-by-Step Solution

Answer

The equation of the common chord is \( x + y + 5 = 0 \).

Step 1: Write the equations of the circles

Consider the circle equations given: 1. \( x^{2} + y^{2} + 4x - 6y - 12 = 0 \)2. \( x^{2} + y^{2} + 8x - 2y + 8 = 0 \)

Step 2: Subtract the two equations

Subtract the first circle's equation from the second circle's equation: \( (x^{2} + y^{2} + 8x - 2y + 8) - (x^{2} + y^{2} + 4x - 6y - 12) = 0 \) Simplify the resulting equation.

Step 3: Simplify the equation

Combine like terms: \( (x^{2} + y^{2}) - (x^{2} + y^{2}) + 8x - 2y + 8 - 4x + 6y + 12 = 0 \) \( 4x + 4y + 20 = 0 \)

Step 4: Solve for the equation of the common chord

Divide by the common factor 4 to simplify the equation: \( x + y + 5 = 0 \)

Key Concepts

Circle Equations

In coordinate geometry, a circle's equation represents all points at a fixed distance (radius) from a center point (h, k). The standard form of a circle equation is \( (x - h)^2 + (y - k)^2 = r^2 \).
Sometimes, the equations of circles are given in expanded form, which needs to be simplified to identify the center and radius.
In this exercise, we are given:
1. Equation 1: \( x^2 + y^2 + 4x - 6y - 12 = 0 \)
2. Equation 2: \( x^2 + y^2 + 8x - 2y + 8 = 0 \)
We need to find the equation of the common chord shared by these circles.
The common chord is where the two circles intersect.

Coordinate Geometry

Coordinate geometry involves the study of geometric figures using the coordinate plane.
The main elements we work with are points, lines, and curves described through algebraic equations.
In problems involving circles, we deal with coordinates that describe the various points forming the circles.
When two circles intersect, their common points form a line, and these points satisfy both circle equations simultaneously.
This line is called the common chord.
To find its equation, we can utilize the provided circle equations.

Equation Simplification

Simplifying equations involves combining like terms and reducing the equation to its simplest form.
In the provided exercise, we start with the expanded forms of two circle equations.
Subtract the first equation from the second to highlight the common chord:
\[ (x^2 + y^2 + 8x - 2y + 8) - (x^2 + y^2 + 4x - 6y -12) = 0 \]
Step-by-step, like terms (i.e., terms containing the same variables with the same powers) are combined:
\[ 4x + 4y + 20 = 0 \]
This resulting equation represents the line of intersection between the two circles in the simplest form.

Subtraction of Equations

Subtraction of equations is a technique in algebra to eliminate variables or to find common solutions.
For the given circle equations, subtract one from the other to find their points of intersection:
1. Start with \( x^2 + y^2 + 8x - 2y + 8 = 0 \)
2. Subtract the equation \[ x^2 + y^2 + 4x - 6y - 12 = 0 \]
This eliminates \( x^2 \) and \( y^2 \) terms as they cancel out:
\[ 4x + 4y + 20 = 0 \]
Finally, divide by the common factor to simplify:
\[ x + y + 5 = 0 \]
This represents the equation of the common chord where the circles intersect.