Problem 101 - Precalculus

Question

The circle given by the equation \(x^{2}+y^{2}+a x+b y+c=0\) passes through the points (4,4) \((-3,-1),\) and \((1,-3) .\) Find \(a, b,\) and \(c\)

Step-by-Step Solution

Answer

\(a = -\frac{22}{17}, b = -\frac{44}{17}, c = -\frac{280}{17}\).

Step 1: Understand the Equation and Points

The given equation of the circle is \(x^{2} + y^{2} + ax + by + c = 0\). We need to use the fact that the circle passes through the points (4,4), (-3,-1), and (1,-3). By substituting each point into this equation, we will get three linear equations to solve for \(a\), \(b\), and \(c\).

Step 2: Substitute the First Point

Substitute the point (4, 4) into the equation:\[4^2 + 4^2 + 4a + 4b + c = 0 \16 + 16 + 4a + 4b + c = 0 \32 + 4a + 4b + c = 0\]Which simplifies to:\[4a + 4b + c = -32\]

Step 3: Substitute the Second Point

Substitute the point (-3, -1) into the equation:\[(-3)^2 + (-1)^2 - 3a - b + c = 0 \9 + 1 - 3a - b + c = 0 \10 - 3a - b + c = 0\]Which simplifies to:\[-3a - b + c = -10\]

Step 4: Substitute the Third Point

Substitute the point (1, -3) into the equation:\[1^2 + (-3)^2 + a - 3b + c = 0 \1 + 9 + a - 3b + c = 0 \10 + a - 3b + c = 0\]Which simplifies to:\[a - 3b + c = -10\]

Step 5: Form the System of Equations

We now have a system of linear equations: 1. \(4a + 4b + c = -32\)2. \(-3a - b + c = -10\)3. \(a - 3b + c = -10\)

Step 6: Solve the System of Equations

To solve the system:Subtract equation 2 from 1:\((4a + 4b + c) - (-3a - b + c) = -32 - (-10)\)\[4a + 4b + c + 3a + b - c = -22\] \[7a + 5b = -22\] Subtract equation 3 from 2:\((-3a - b + c) - (a - 3b + c) = -10 - (-10)\)\[-3a - b - a + 3b = 0\] \[-4a + 2b = 0\] Which boils down to:\[2a = b\]

Step 7: Substitute and Solve for Variables

Using \[2a = b\], substitute into \[7a + 5b = -22\]:\[7a + 5(2a) = -22\]\[7a + 10a = -22\]\[17a = -22\]\[a = -\frac{22}{17}\]Then \[b = 2a = 2\left(-\frac{22}{17}\right) = -\frac{44}{17}\].Finally, substitute \(a\) and \(b\) into one of the previous equations to find \(c\):Substitute into \[4a + 4b + c = -32\]:\[4(-\frac{22}{17}) + 4(-\frac{44}{17}) + c = -32\]\[-\frac{88}{17} - \frac{176}{17} + c = -32\]\[-\frac{264}{17} + c = -32\]\[c = -32 + \frac{264}{17}\]\[c = \frac{-544 + 264}{17}\]\[c = \frac{-280}{17}\]

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. In the context of this exercise, we are presented with a situation where a circle, given by the equation \(x^2 + y^2 + ax + by + c = 0\), passes through three specific points. By substituting these points into the equation, we derived three separate linear equations. Each equation represents a relationship between the variables \(a\), \(b\), and \(c\), which define the circle's position and size.

Here are the linear equations from our problem:

• \(4a + 4b + c = -32\)
• \(-3a - b + c = -10\)
• \(a - 3b + c = -10\)

Solving a system of linear equations usually involves finding a common solution for all the variables. In this problem, that means determining the values of \(a\), \(b\), and \(c\) so that all three equations are satisfied simultaneously. This process often involves methods like substitution or elimination to simplify and solve the equations.

Substitution Method

The substitution method is a technique used to solve systems of equations. It involves isolating one variable in one of the equations and then substituting that expression into the other equations. This method is particularly useful when dealing with equations that have clear substitutions.

In this exercise, to find \(a\), \(b\), and \(c\), we first determined a simpler relationship between \(a\) and \(b\) from one of the equations by elimination. Specifically, from equations \(-3a - b + c = -10\) and \(a - 3b + c = -10\), we derived the equation \(2a = b\).

This expression \(b = 2a\) is then substituted back into the other equations to find the values for \(a\) and subsequently \(b\). Once \(a\) and \(b\) are found, they are substituted into any original equation to find \(c\). This step-by-step substitution helps in reducing the complexity and solving the system efficiently.

• Substitute \(b = 2a\) into \(7a + 5b = -22\) to find \(a\).
• Back-substitute \(a\) to get \(b\).
• Substitute both into \(4a + 4b + c = -32\) to find \(c\).

Analytic Geometry

Analytic geometry, also known as coordinate geometry, is the study of geometry using a coordinate system. This branch of mathematics allows the translation of geometric problems into algebraic equations, which are easier to manipulate and solve.

In this problem, the circle is represented analytically using the equation:
\(x^2 + y^2 + ax + by + c = 0\). Instead of a traditional geometric representation, this exercise uses the circle's algebraic form to compute specific parameters (\(a\), \(b\), and \(c\)).

The beauty of analytic geometry lies in its power to convert geometric conditions into equations that can be solved using algebraic techniques. In our case, we used the fact that the circle must satisfy the condition of passing through certain points, leading us to solve a system of linear equations to find the desired parameters. Through this method, complex geometric problems become solvable with the application of algebra.