Finding the Equation of a Circle Passing Through Given Points

Understanding how to determine the equation of a circle that passes through specific points is a crucial skill in algebra and geometry. In this article, we will walk through the process of finding the equation of a circle that goes through the points (0,1), (1,4), and (5,2).

Introduction to the General Equation of a Circle

The general equation of a circle in the Cartesian plane is given by:

x^2 y^2 Dx Ey F 0

Where D,E, and F are constants that need to be determined.

Finding the Constants

To find the constants, we will substitute the given points into the general equation of a circle and solve the resulting system of equations.

Step 1: Setting Up the System of Equations

Given points are (0,1), (1,4), and (5,2). Substituting these into the general equation, we create the following system of equations:

0^2 1^2 D(0) E(1) F 0 Rightarrow 1 E F 0 Rightarrow E F -1 quad (Equation 1) 1^2 4^2 D(1) E(4) F 0 Rightarrow 1 16 D 4E F 0 Rightarrow D 4E F -17 quad (Equation 2) 5^2 2^2 D(5) E(2) F 0 Rightarrow 25 4 5D 2E F 0 Rightarrow 5D 2E F -29 quad (Equation 3)

Step 2: Solving the System of Equations

From the above system of equations, we start by expressing F from Equation 1:

From Equation 1: F -1 - E

Substituting F in Equations 2 and 3:

D 4E - 1 - E -17 Rightarrow D 3E - 1 -17 Rightarrow D 3E -16 quad (Equation 4) 5D 2E - 1 - E -29 Rightarrow 5D E - 1 -29 Rightarrow 5D E -28 quad (Equation 5)

Step 3: Solving Equations 4 and 5

From Equation 4, we can express D in terms of E:

D -16 - 3E

Substituting this into Equation 5:

5(-16 - 3E) E -28 Rightarrow -80 - 15E E -28 Rightarrow -80 - 14E -28 Rightarrow -14E 52 Rightarrow E -frac{52}{14} -frac{26}{7}

Substituting E -frac{26}{7} back into Equation 4 to find D:

D - frac{78}{7} -16 Rightarrow D -16 frac{78}{7} -frac{112}{7} frac{78}{7} -frac{34}{7}

Substituting E -frac{26}{7} back into Equation 1 to find F:

F -1 - left( -frac{26}{7} right) -1 frac{26}{7} -frac{7}{7} frac{26}{7} frac{19}{7}

Step 4: Writing the Circle Equation

Thus, we have:

D -frac{34}{7} E -frac{26}{7} F frac{19}{7}

The equation of the circle is:

x^2 y^2 - frac{34}{7}x - frac{26}{7}y frac{19}{7} 0

Multiplying through by 7 to clear fractions:

7x^2 7y^2 - 34x - 26y - 19 0

Final Equation