Equation of a Circle when the Center of Circle Coincides with the Origin

In this tutorial we will learn, equation of a circle when the center of circle coincides with the origin with examples.

Equation of a circle with center at (h, k) and radius equal to r, is

(x – h)² + (y – k)² = r²……(1)

When the center of the circle(h, k) is (0, 0), coincides with the origin so, h = 0 and k = 0.

Putting the value in eq.(1)

(x – h)² + (y – k)² = r²

(x – 0)² + (y – 0)² = r²

Hence required equation of the circle is x² + y² = r².

Consider an arbitrary point P(x, y) on a circle. Let ‘r’ be the radius of the circle which equal to OP.

We know that the distance between the point (x, y) and origin (0, 0) can be found using distance formula which is equal to √(x² + y²) = r.

Therefore, the equation of a circle, with the center as origin is x² + y² = r².

Alternative Method

Let (x, y) is a point on a circle and the center of the circle is at origin (0, 0).

Now we draw a perpendicular from point (x, y) to the x-axis, then we get a right angle triangle. In this right angle triangle radius of the circle is the hypotenuse.

The base of the triangle is the distance along x-axis and height is the distance along the y-axis.

Thus applying the Pythagoras theorem, we get

x² + y² = r²

Equation of a circle when the center is origin is x² + y² = r²

Example: Find the equation of the circle whose center coincides with the origin and radius is 4 cm.

Solution: The equation of a circle whose center coincides with the origin and radius is 4 cm is

x² + y² = r²

x² + y² = 4²

x² + y² – 16 = 0

Example: Find the equation of the circle whose center coincides with the origin and radius is 2 cm.

Solution: The equation of a circle whose center coincides with the origin and radius is 2 cm is

x² + y² = r²

x² + y² = 2²

x² + y² = 4

x² + y² – 4 = 0

Example: Find the equation of the circle whose center coincides with the origin and radius is 9 cm.

Solution: The equation of a circle whose center coincides with the origin and radius is 9 cm is

x² + y² = r²

x² + y² = 9²

x² + y² = 81

x² + y² – 81 = 0