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How to Convert to Center-Radius Form for a Circle?

Published in Uncategorized 4 mins read

The center-radius form, also known as the standard form of a circle's equation, is a fundamental way to represent a circle. It allows for immediate identification of the circle's center coordinates and its radius, making it incredibly useful for graphing and analyzing circles. While the question asks "How to do center radius form," it generally refers to the process of converting a circle's equation from its general or standard form into the center-radius form, which reveals its key properties.

What is the Center-Radius Form of a Circle?

The center-radius form of a circle's equation is expressed as:

$(x - h)^2 + (y - k)^2 = r^2$

Where:

  • $(h, k)$ represents the coordinates of the circle's center.
  • $r$ represents the length of the circle's radius.
  • $x$ and $y$ are the variables for any point $(x,y)$ on the circle.

Why Convert to Center-Radius Form?

Converting an equation of a circle from its general form (e.g., $Ax^2 + Ay^2 + Bx + Cy + D = 0$) to the center-radius form offers significant advantages:

  • Immediate Identification: You can directly read the center $(h,k)$ and the radius $r$ from the equation, simplifying analysis.
  • Easy Graphing: Knowing the center and radius simplifies plotting the circle accurately on a coordinate plane.
  • Problem Solving: It streamlines various geometric problems involving circles, such as finding distances or intersections.

Steps to Convert from General Form to Center-Radius Form

The primary method for converting a circle's equation from general form ($Ax^2 + Ay^2 + Bx + Cy + D = 0$) to center-radius form involves a technique called completing the square.

Here's a step-by-step guide:

  1. Rearrange and Group Terms:

    • Move the constant term ($D$) to the right side of the equation.
    • Group the $x$-terms together and the $y$-terms together on the left side, leaving space for the terms you'll add to complete the square.
    • Example: If you start with $x^2 + y^2 + Bx + Cy + D = 0$, it becomes $(x^2 + Bx) + (y^2 + Cy) = -D$.
  2. Ensure Leading Coefficients are 1:

    • If the coefficients of $x^2$ and $y^2$ (represented by $A$) are not 1, divide the entire equation by $A$. (For an equation to represent a circle, the coefficients of $x^2$ and $y^2$ must be equal and non-zero).
  3. Complete the Square for the X-terms:

    • Take the coefficient of your $x$-term (let's say $B$), divide it by 2, and then square the result: $(\frac{B}{2})^2$.
    • Add this calculated value to both sides of the equation.
  4. Complete the Square for the Y-terms:

    • Similarly, take the coefficient of your $y$-term (let's say $C$), divide it by 2, and then square the result: $(\frac{C}{2})^2$.
    • Add this value to both sides of the equation.
    • As highlighted in the video reference, "In a circle the Y terms are also squared. So we complete the square twice" – once for the X-terms and once for the Y-terms. This differs from completing the square for a standard quadratic which only involves one squared variable.
  5. Factor the Perfect Square Trinomials:

    • The expressions you created on the left side (e.g., $x^2 + Bx + (\frac{B}{2})^2$ and $y^2 + Cy + (\frac{C}{2})^2$) are now perfect square trinomials.
    • Factor them into the form $(x - h)^2$ and $(y - k)^2$. Specifically, these will be $(x + \frac{B}{2})^2$ and $(y + \frac{C}{2})^2$. Remember that $h$ and $k$ in the center $(h,k)$ will take the opposite sign of the numbers inside the parentheses.
  6. Simplify the Right Side:

    • Add all the numbers on the right side of the equation. This sum will be your $r^2$. If the value on the right side is negative, the equation does not represent a real circle.
  7. Identify Center and Radius:

    • Once the equation is in the form $(x - h)^2 + (y - k)^2 = r^2$, the center is $(h, k)$ and the radius is $r = \sqrt{\text{value on the right side}}$.

Example Conversion

Let's convert the general form equation $x^2 + y^2 - 6x + 10y + 18 = 0$ to center-radius form.

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- 2:48

  • 8:58

And once for Y. Alright. So that's the difference between completing. The square for a quadratic. More And once for Y. Alright. So that's the difference between completing. The square for a quadratic. And for a circle. In a circle the Y terms are also squared. So we complete the square twice.