Circle with Centre (0,0)

What is a Circle?

A circle is the set of all points in a plane that are at a fixed distance, called the radius ( $r$ ), from a fixed point known as the centre.

If the centre of the circle is at the origin $(0, 0)$ , and the radius is $r$ , the equation of the circle is:

x^2 + y^2 = r^2

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Problem: Is the point $(3, 4)$ on the circle with equation $x^2 + y^2 = 25$ ?

Solution:

Substitute $(x, y) = (3, 4)$ into $x^2 + y^2 = 25$ :

3^2 + 4^2 = 9 + 16 = 25

The equation is satisfied, so $(3, 4)$ lies on the circle.

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Problem: A circle has the equation $x^2 + y^2 = 49$ .

Find the radius.

Solution:

Compare with the standard form $x^2 + y^2 = r^2$ :

r^2 = 49 \quad \Rightarrow \quad r = \sqrt{49} = 7

Answer: The radius is 7.

Definition: A circle is the set of all points equidistant from a fixed centre.
Standard Equation: $x^2 + y^2 = r^2$ $x^{2} + y^{2} = r^{2}$
- $r$ is the radius.
Key Properties:
- Radius = $r$ , Diameter = $2r$
- Circumference = $2\pi r$
- Area = $\pi r^2$
Use the equation to verify points or find the radius of a circle.