Circle Equation Calculator Form
How Circle Equation Calculator Works
- You enter the center of the circle, \( (h, k) \).
- You select whether to input the radius \( r \) or a point \( (x, y) \) that lies on the circle. A radio button allows you to toggle between these two options.
- If you select the "Radius" option, you enter the radius \( r \) in the provided field. The point input fields will be disabled in this case.
- If you select the "Point" option, you enter the point \( (x, y) \) on the circle, and the calculator automatically calculates the radius using the formula: \[ r = \sqrt{(x - h)^2 + (y - k)^2} \]
- The calculator validates your input. If you provide invalid values, like non-numeric characters or a negative radius, it prompts you to enter valid values.
- Once you enter valid inputs, the calculator constructs the equation of the circle in the standard form: \[ (x - h)^2 + (y - k)^2 = r^2 \] It then displays this equation as the final output.
Understanding the Equation of a Circle
A circle is a set of points in a plane that are all equidistant from a fixed point known as the center. The standard form of a circle's equation is derived using the center and the radius of the circle.
The Standard Form of a Circle's Equation
The standard form of the equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Here:
- \( (h, k) \) is the center of the circle
- \( r \) is the radius of the circle
- \( (x, y) \) represents any point on the circle
Derivation for the Equation of Circle
To derive this equation, consider a point \( (x, y) \) that lies on the circle with center \( (h, k) \) and radius \( r \). The distance between the center and the point on the circle is always equal to the radius. This relationship can be expressed using the distance formula:
\[ \sqrt{(x - h)^2 + (y - k)^2} = r \]
By squaring both sides of the equation, we eliminate the square root and arrive at the standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Examples
Example 1: Calculate the Equation of a Circle whose center is at (2, -3) and a Radius of 5 units.
We will find the equation of a circle with its center at \( (h, k) = (2, -3) \) and a radius \( r = 5 \).
Step 1: Identify the center and radius
From the problem, we know: \[ h = 2, \quad k = -3, \quad r = 5 \] The center is \( (2, -3) \), and the radius is 5 units.
Step 2: Write the standard form of the circle equation
The standard form of a circle’s equation is: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substitute the values for \( h \), \( k \), and \( r \): \[ (x - 2)^2 + (y + 3)^2 = 5^2 \]
Step 3: Simplify the equation
Simplify the radius term: \[ (x - 2)^2 + (y + 3)^2 = 25 \] This is the equation of the circle with center \( (2, -3) \) and radius \( 5 \).
Example 2: Calculating the Equation of a Circle Passing Through a Given Point
We will find the equation of a circle with its center at \( (h, k) = (-1, 4) \) and passing through the point \( (3, 1) \).
Step 1: Identify the center and point on the circle
From the problem, we know: \[ h = -1, \quad k = 4, \quad \text{Point on circle} = (3, 1) \] The center is \( (-1, 4) \), and the circle passes through the point \( (3, 1) \).
Step 2: Calculate the radius using the distance formula
To find the radius, we use the distance formula between the center \( (h, k) \) and the point on the circle \( (x_1, y_1) \): \[ r = \sqrt{(x_1 - h)^2 + (y_1 - k)^2} \] Substitute the values:
\[ r = \sqrt{(3 - (-1))^2 + (1 - 4)^2} \]
\[ r = \sqrt{(3 + 1)^2 + (1 - 4)^2} \]
\[ r = \sqrt{4^2 + (-3)^2} \]
\[ r = \sqrt{16 + 9} = \sqrt{25} \]
\[ r = 5 \]
Thus, the radius is 5 units.
Step 3: Write the standard form of the circle equation
Now that we know the radius is 5, we use the standard form of a circle’s equation: \[ (x - h)^2 + (y - k)^2 = r^2 \] Substitute \( h = -1 \), \( k = 4 \), and \( r = 5 \): \[ (x + 1)^2 + (y - 4)^2 = 5^2 \]
Step 4: Simplify the equation
Simplify the radius term: \[ (x + 1)^2 + (y - 4)^2 = 25 \] This is the equation of the circle with center \( (-1, 4) \) and radius 5, which passes through the point \( (3, 1) \).