Hello, young mathematicians! Today, we’re going to learn how to determine if a given equation represents a circle. To make this easy to understand, we’ll use an example and go through the steps together.
Example 2: Determine if the Equation is a Circle
Let’s find out if the equation x² + y² + 2x + 2y + 2 = 0 represents a circle.
Steps to Solve:
Step 1: Check the Coefficients of Squared Terms and Simplify the Equation
If the coefficients of x² and y² are not 1, we divide each term by that coefficient. In our example, both coefficients are 1, so we can skip this step.
Step 2: Group the Terms and Move the Constant
Group the x-terms and y-terms together and move the constant to the right side of the equation.
x² + y² + 2x + 2y + 2 = 0
(x² + 2x) + (y² + 2y) = – 2 ———(1)
Step 3: Complete the Square
To complete the square, divide the coefficient of first-degree term by 2 and then square it.
- For (x² + 2x),
2x is first-degree term. 2 is co-efficient of x.
- For (y² + 2y),
2y is first-degree term. 2 is co-efficient of y.
- 2 ÷ 2 = 1
- 1² = 1
Step 4: Add the Squared Terms
Add these squared terms within their respective groups and to the right side of the equation to maintain equality.
(x² + 2x) + (y² + 2y) = – 2 ———(1)
(x² + 2x + 1) + (y² + 2y + 1) = – 2 + 1 + 1
(x² + 2x + 1) + (y² + 2y + 1) = 0 ———-(2)
Step 5: Write as a Perfect Square
We know the formulae, if a and b are two variables then:
a² + 2ab + b² = (a + b)² ——————– (A)
a² – 2ab + b² = (a – b)² ——————— (B)
Applying this formula means writing as a perfect square.
Here,
(x² + 2x + 1) = (x + 1)²
(y² + 2y + 1) = (y + 1)²
Thus equation (2) becomes
(x² + 2x + 1) + (y² + 2y + 1) = 0 ———-(2)
(x + 1)² + (y + 1)² = 0
Step 6: Identify the Circle
(x + 1)² + (y + 1)² = 0
This is not the standard equation of a circle because the right side of the equation represents the radius squared r² of the circle, and it cannot be zero. Instead, it represents a point at (-1, -1).
Therefore, the given equation x² + y² + 2x + 2y + 2 = 0 represents a point, not a circle.
Summary
To determine if an equation is a circle, follow these steps:
- Simplify the equation by dividing all terms if necessary.
- Group the x-terms and y-terms.
- Complete the square for the x-terms and y-terms.
- Add the squared terms.
- Write as a perfect square.
- Determine if the equation represents a circle in standard form. If it does, extract the center and radius from the equation.
By following these steps, you can find out if any given equation represents a circle. Keep practicing, and you’ll get the hang of it!
I hope this post helps you understand how to determine if an equation represents a circle. Keep practicing and have fun!