Problem 73 - Calculus Single Variable

Question

Find the equation of the circle that passes through the points \((2,4),(4,0),\) and (-5,-3)

Step-by-Step Solution

Answer

The equation of the circle is \(x^2 + y^2 - 2x - 2y - 8 = 0\).

Step 1: General Equation of a Circle

The general equation of a circle is \(x^2 + y^2 + Dx + Ey + F = 0\). Our task is to find the coefficients \(D\), \(E\), and \(F\).

Step 2: Substitute Points into the Equation

By substituting the points \((2,4)\), \((4,0)\), and \((-5,-3)\) into the general equation, we'll obtain a system of equations. For (2,4): \[2^2 + 4^2 + 2D + 4E + F = 0\]leads to:\[ 4 + 16 + 2D + 4E + F = 0 \] or \[ 2D + 4E + F = -20 \] For (4,0): \[4^2 + 0^2 + 4D + 0E + F = 0\]leads to:\[ 16 + 4D + F = 0 \] or \[ 4D + F = -16 \]For (-5,-3):\[(-5)^2 + (-3)^2 + (-5)D + (-3)E + F = 0\]leads to:\[ 25 + 9 - 5D - 3E + F = 0 \] or \[ -5D - 3E + F = -34 \].

Step 3: Solve the System of Equations

We now have the system of equations:1. \(2D + 4E + F = -20\)2. \(4D + F = -16\)3. \(-5D - 3E + F = -34\)First, solve equations 2 and 3 for \(F\) and substitute:From (2): \(F = -4D - 16\)Substitute into equation (1):\[ 2D + 4E + (-4D - 16) = -20 \]\[-2D + 4E = -4 \]\[ E = rac{2D - 4}{4} \] or \[ E = rac{D - 2}{2} \]Now substitute \(E\) in equation 3:\[-5D - 3\left(\frac{D-2}{2}\right) + (-4D - 16) = -34\]Solve for \(D\) to get \(D = -2\).Then substitute \(D\) back to find \(E\):\[E = rac{-2 - 2}{2} = -2\]. Finally substitute \(D\) into the equation for \(F\):\[F = -4(-2) - 16 = -8\].

Step 4: Write the Equation of the Circle

Now that we have \(D = -2\), \(E = -2\), and \(F = -8\), substitute these back into the general equation to get the specific circle equation: \[x^2 + y^2 - 2x - 2y - 8 = 0\].

Key Concepts

General Equation of a Circle

The general equation of a circle is an essential tool in coordinate geometry. It provides a way to describe every point that forms a circle in the Cartesian plane. This equation is expressed as: \[x^2 + y^2 + Dx + Ey + F = 0\] In this equation, \(D\), \(E\), and \(F\) are constants that need to be determined. Here,

• \(x^2 + y^2\) represents a perfect circle centered at the origin.
• \(Dx + Ey\) shifts the circle's center in the Cartesian plane.
• \(F\) adjusts the radius.

To find a specific circle's equation, like one passing through given points, you must determine these constants, often through substituting known coordinates into the equation. This provides you with a system of equations to solve for \(D\), \(E\), and \(F\). Take note that solving these coefficients properly connects all parts of the circle equation.

System of Equations

When you're working through problems involving the general equation of a circle, you often encounter a system of equations. This happens because you have to substitute several points into that general equation to find the unknowns. Take this problem as an example:

• The solution method started by substituting the given points, like \((2,4)\), into the general circle equation.
• This substitution results in equations based on each set of coordinates.
• Using all these equations together forms a system of equations.

Effectively solving a problem means managing these simultaneous equations through substitution or elimination, allowing you to find the values of your unknowns. It might initially seem challenging, but tackling these equations step by step helps illuminate each part of the problem.

Solving Equations Step by Step

Breaking down any problem into simple, understandable steps is crucial. It makes challenging tasks manageable and ensures accuracy. Here's a simple approach you could follow when solving equations like this: - **Identify each part:** Begin by clearly understanding the equation's structure, like the general form of a circle. - **Substitute precisely:** Use given points to substitute into the equation accurately. - **Simplify equations:** For each substitution, simplify as much as possible, leaving clear, straightforward forms. - **Manage systematically:** With your system of equations at hand, use methods like substitution or elimination to solve step by step. Careful attention to each of these steps ensures no errors creep into your calculations, paving the way to a correct and clear final solution.

Coordinate Geometry

Coordinate geometry is a powerful mathematical tool, allowing us to solve problems involving figures like circles using algebra. By working in a coordinated plane, you can: - **Visualize geometric figures:** Understand their position and dimension by working with directional axes (\(x\)- and \(y\)-axes). - **Analyze relationships:** Find relationships between different points using distance formulas or by substituting into general equations. - **Determine specific attributes:** For circles, use parameters like radius or center in your equations. Understanding coordinate geometry means you see each geometric figure as part of a larger mathematical landscape. Solving equations in this domain strengthens your ability to predict shapes and paths of figures, essential skills in more advanced geometry applications.