Problem 8 - The Calculus with Analytic Geometry

Question

In Exercises 5 through 10, find an equation of the circle satisfying the given conditions. Through the three points \((2,8),(7,3)\), and \((-2,0)\).

Step-by-Step Solution

Answer

The equation of the circle is \( x^2 + (y - 1)^2 = 53 \).

Step 1: Recall the general form of a circle

The general form of the equation of a circle with center \( h, k \) and radius \( r \) is \( (x - h)^2 + (y - k)^2 = r^2 \).

Step 2: Set up the system of equations

Substitute the given points \( (2,8) \), \( (7,3) \), and \( (-2,0) \) into the circle equation. This will form three equations: \( (2 - h)^2 + (8 - k)^2 = r^2 \), \( (7 - h)^2 + (3 - k)^2 = r^2 \), \( (-2 - h)^2 + (0 - k)^2 = r^2 \).

Step 3: Expand and simplify the equations

Expand each equation: 1) \( (2 - h)^2 + (8 - k)^2 = r^2 \) simplifies to \( 4 - 4h + h^2 + 64 - 16k + k^2 = r^2 \), or \( h^2 + k^2 - 4h - 16k + 68 = r^2 \). 2) \( (7 - h)^2 + (3 - k)^2 = r^2 \) simplifies to \( 49 - 14h + h^2 + 9 - 6k + k^2 = r^2 \), or \( h^2 + k^2 - 14h - 6k + 58 = r^2 \). 3) \( (-2 - h)^2 + (0 - k)^2 = r^2 \) simplifies to \( 4 + 4h + h^2 + k^2 = r^2 \), or \( h^2 + k^2 + 4h + k^2 + 4 = r^2 \).

Step 4: Set up systems of linear equations

Form equations by subtracting one simplified equation from another to eliminate \( r^2 \). This will give: (i) \( 10h + 10k = 10 \), simplified to \( h + k = 1 \). (ii) \( -18h + 10k = 10 \), simplified to \( h + 2k = 2 \).

Step 5: Solve for the center \( h, k \)

Now, solve the system of equations: From \( h + k = 1 \), and \( h + 2k = 2 \), we get \( k = 1 \) and \( h = 0 \). So, the center of the circle is \( (0,1) \).

Step 6: Solve for the radius \( r \)

Substitute \( h = 0 \) and \( k = 1 \) back into one of the original circle equations: \( (2 - 0)^2 + (8 - 1)^2 = r^2 \), leading to \( 4 + 49 = r^2 \), so \( r^2 = 53 \). Thus, \( r = \sqrt{53} \).

Step 7: Write the final equation of the circle

The equation of the circle with center \( (0,1) \) and radius \( \sqrt{53} \) is: \( (x - 0)^2 + (y - 1)^2 = 53 \), which simplifies to \( x^2 + (y - 1)^2 = 53 \).

Key Concepts

Circle Center

The center of a circle is a crucial concept in defining the geometry of the circle. It is represented by the coordinates \( (h, k) \). The center is the fixed point from which all points on the circle are equidistant (the radius). To find the center of a circle given three points, you need to set up a system of equations derived from the general circle equation: \( (x - h)^2 + (y - k)^2 = r^2 \). For each point, substituting its coordinates into the equation gives three equations with three unknowns (\

System of Linear Equations

Solving multiple equations in multiple variables often involves using a system of linear equations. In the context of finding the circle's equation, after substituting each point into the circle equation, you get a system of linear equations. By eliminating \( r^2 \) through subtraction, you derive simpler equations: \( 10h + 10k = 10 \) simplified to \( h + k = 1 \), and \( -18h + 10k = 10 \) simplified to \( h + 2k = 2 \). Solving this system provides the circle's center coordinates. Apply substitution or elimination methods to find the unique solution for \( h \) and \( k \).