## Equation of a CircleIn geometry, a In this section, we will learn the There are two forms of the equation of a circle: - Standard Form
- General Form
## Standard FormIf the equation of a circle is in standard form, we can easily find the center of the circle (h, k) and the radius of the circle. The standard equation of a circle is: (x-h) ^{2}+(y-k)^{2}=r^{2}Where Let's see some examples based on the standard form.
The given equation is,
Compare the given equation with the standard form, we get: h=2,k=3 and r Now, we can plot the circle on the graph paper with radius r = 2 and center (2, 3).
The given equation is, The above equation is not matching with the standard form. So, first, we convert the equation in the standard form by dividing the equation by 2. Solving the above equation, we get: x We can write the above equation as: (x-0) Compare the above equation with the standard form, we get: h=0,k=0 and r Now, we can plot the circle on the graph paper with radius r = 2 and center (0, 0).
The given equation is, Compare the given equation with the standard form, we get: h=4,k=-5 We see that y coordinate is negative. In general, y term is (y-k)
r
Given, radius (r) = 25 cm Center coordinate (h, k) = (-2, 6) (x-(-2))
In the given figure, the center coordinates (h, k) are (0, 0) and the radius (r) is 4. Hence, the equation of the circle is:
## General FormThe general form of the equation is the expanded form of the standard equation. We know the standard equation of the circle:
Expanding the equation (1), we get: x Rearrange the above equation, we get: x Substitute the values of h, k, and r by the following values, we get: h=-g,k=-f,c=h Put these values in the equation (2), we get: x Where (-g, -f) is the center of the circle, and the radius (r) is √g x ^{2}+y^{2}+2gx+2fy+c=0Where g, f, and c are constants. We can further substitute the values 2g, 2f, and c by x ^{2}+y^{2}+Dx+Ey+F=0Where D, E, and F are constants.
- If the value of
**g**, the radius of the circle is real and the equation represents a^{2}+f^{2}-c^{2}>0**real** - If the value of
**g**, the radius of the circle also becomes 0. In the particular case, the circle reduces to the point (-g, -f) and the circle is called a^{2}+f^{2}-c^{2}=0**point circle**. - If the value of
**g**, the radius of the circle becomes imaginary. But the circle is not imaginary, it is real. This type of circle is called an^{2}+f^{2}-c^{2}<0**imaginary circle**.
Let's solve some examples based on general form.
The given equation is First, we will divide the whole equation by 4, we get: Rearranging the above equation, we get: x Now we will find the value of g and f, respectively. We know that, In the above equation, the coefficient of x is -4. Therefore, Similarly, we will find the value of f. We know that, In the above equation, the coefficient of y is 6. Therefore, ## Note: We add the value of g |