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Magazine Chapter 10: Problem 90
Find an equation of the circle that passes through the points \((2,3),(4,5),\) and \((0,-3)\)
Short Answer
Expert verified
The equation of the circle passing through the points (2,3), (4,5), and (0,-3) is \[ x^2 + y^2 - 21x - 7y - 30 = 0 \].
Step by step solution
01
Determine the general equation of the circle
The general equation of a circle in Cartesian coordinates is given by \[ x^2 + y^2 + Dx + Ey + F = 0 \]. Here, we will find the constants D, E, and F.
02
Substitute point \(2,3\) into the equation
Substitute \( x = 2 \) and \( y = 3 \) into the general equation to get one equation: \[ 2^2 + 3^2 + 2D + 3E + F = 0 \] which simplifies to \[ 4 + 9 + 2D + 3E + F = 0 \] or \[ 2D + 3E + F = -13 \] \eqref{eq1}
03
Substitute point \(4,5\) into the equation
Substitute \( x = 4 \) and \( y = 5 \) into the general equation to get a second equation: \[ 4^2 + 5^2 + 4D + 5E + F = 0 \] which simplifies to \[ 16 + 25 + 4D + 5E + F = 0 \] or \[ 4D + 5E + F = -41 \] \eqref{eq2}
04
Substitute point \(0,-3\) into the equation
Substitute \( x = 0 \) and \( y = -3 \) into the general equation to get a third equation: \[ 0^2 + (-3)^2 + 0D - 3E + F = 0 \] which simplifies to \[ 9 - 3E + F = 0 \] or \[ -3E + F = -9 \] \eqref{eq3}
05
Solve the system of equations
Use the equations obtained to solve for D, E, and F: \[ 2D + 3E + F = -13 \] \eqref{eq1}\[ 4D + 5E + F = -41 \] \eqref{eq2}\[ -3E + F = -9 \] \eqref{eq3}Subtract eq\eqref{eq1} from eq\eqref{eq2}: \[ 2D + 2E = -28 \] which simplifies to \[ D + E = -14 \] \eqref{eq4} Now solve eq\eqref{eq4} for D: \[ D = -14 - E \] \eqref{eq5}Substitute eq\eqref{eq5} into eq\eqref{eq3} to solve for E and F.
06
Find values for D, E, and F
Substitute eq5 into eq3, \[ -3E + F = -9 \] simplifies to\[ -3E + ( -9-5E)= -9\] to solve for E, F: \[ E = -7\]\[D= -21\]\[ F= -30\]
07
Write the final equation of the circle
Substitute D, E, and F back into the general equation: \[ x^2 + y^2 - 21x - 7y - 30 = 0 \] This is the equation of the circle passing through the points (2,3), (4,5), and (0,-3).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
general equation of a circle
A circle in the Cartesian coordinate system can be represented by an equation. The general equation of a circle is written as: \[ x^2 + y^2 + Dx + Ey + F = 0 \]where
- D, E, and F are constants that need to be determined.
- points through which the circle passes
- the center and radius of the circle
system of equations
To determine the constants D, E, and F, you need to form a system of equations. This system is built by substituting the given points into the general equation of the circle. When you substitute a point into the general equation, you get a new equation. For each point, you create one equation:
- Substitute (2,3): \[ 2^2 + 3^2 + 2D + 3E + F = 0 \]This simplifies to \[ 2D + 3E + F = -13 \]
- Substitute (4,5): \[ 4^2 + 5^2 + 4D + 5E + F = 0 \]This simplifies to \[ 4D + 5E + F = -41 \]
- Substitute (0,-3): \[ 0^2 + (-3)^2 + 0D - 3E + F = 0 \]This simplifies to \[ -3E + F = -9 \]
substitution method
The substitution method is a way to solve a system of equations. It involves solving one equation for one variable, then substituting that result into another equation. Let's solve step-by-step:
- We start with: \[ 2D + 3E + F = -13 \]and \[ 4D + 5E + F = -41 \]Subtract the first equation from the second to find: \[ 2D + 2E = -28 \]
- This simplifies to: \[ D + E = -14 \]
- Rearrange to solve for D: \[ D = -14 - E \]
- Substitute this back into the third equation: \[ -3E + (-14 - E) = -9 \]Solving this, we find: E = -7
- Using E = -7, substitute back to find:\[ D = -21\]
- Lastly, use these values in any original equation to solve for F: \[ F = -30 \]
coordinate geometry
Coordinate geometry allows you to represent geometrical shapes algebraically using a coordinate system. In this exercise, we used coordinate geometry to find the equation of a circle.
- We started with: three given points.Using the general equation of the circle, we created a system of equations.
- We applied the substitution method to solve this system, finding specific constants.
- Finally, we substituted these constants back into the general equation of the circle. This resulted in:\[ x^2 + y^2 - 21x - 7y - 30 = 0 \]This equation represents the circle passing through the given points.Understanding coordinate geometry helps in solving various real-world problems involving shapes and their properties.
