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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 3: Determine if the Equation is a Circle
Let’s find out if the equation x² + y² + 10y + 26 = 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 + 10y + 26 = 0
x² + (y² + 10y) = – 26 ———(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,
There is no first-degree term of x. So, leave it like this.
- For y² + 10y,
10y is first-degree term. 10 is co-efficient of y.
- 10 ÷ 2 = 5
- 5² = 25
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 + (y² + 10y) = – 26 ———(1)
x² + (y² + 10y + 25) = – 26 + 25
x² + (y² + 10y + 25) = – 1 ———-(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,
y² + 10y + 25 = (y + 5)²
Thus equation (2) becomes
x² + (y² + 10y + 25) = – 1 ———-(2)
x² + (y + 5)² = -1
Step 6: Identify the Circle
(x – 0)² + {y – (-5)}² = -1
In Standard Equation of Circle the constant on right hand side of equal sign represents radius² of that circle. Here, we are getting -1.
radius² = -1
radius = √-1
The square root of a negative number is undefined.
Thus, the given equation x² + y² + 10y + 26 = 0 doesn’t have any graph.
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!