# Circle Touches both x-axis and y-axis

We will learn how to find the equation of a circle touches both x-axis and y-axis.

The equation of a circle with centre at (h, k) and radius equal to a, is (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$.

When the circle touches both x-axis and y-axis i.e., h = k = a.

Then the equation (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ becomes (x - a)$$^{2}$$ + (y - a)$$^{2}$$ = a$$^{2}$$

If a circle touches both the co-ordinate axes then the abscissa as well as ordinate of the centre will be equal to the radius of the circle. Hence, the equation of the circle will be of the form:

(x - a)$$^{2}$$ + (y - a)$$^{2}$$ = a$$^{2}$$

⇒ x$$^{2}$$ + y$$^{2}$$ - 2ax - 2ay + a$$^{2}$$ = 0

Solved example on the central form of the equation of a circle touches both x-axis and y-axis:

1. Find the equation of a circle whose radius is 4 units and touches both x-axis and y-axis.

Solution:

Radius of the circle = 4 units.

Since, the circle touches both x-axis and y-axis the centre of the circle is (4, 4).

The required equation of the circle whose radius is 4 units and touches both x-axis and y-axis is

(x - 4)$$^{2}$$ + (y - 4)$$^{2}$$ = 4$$^{2}$$

⇒ x$$^{2}$$ - 8x + 16 + y$$^{2}$$ - 8y + 16 = 16

⇒ x$$^{2}$$ - 8x - 8y + 16 = 0

2. Find the equation of a circle whose radius is 8 units and touches both x-axis and y-axis.

Solution:

Radius of the circle = 8 units.

Since, the circle touches both x-axis and y-axis the centre of the circle is (8, 8).

The required equation of the circle whose radius is 8 units and touches both x-axis and y-axis is

(x - 8)$$^{2}$$ + (y - 8)$$^{2}$$ = 8$$^{2}$$

⇒ x$$^{2}$$ - 16x + 64 + y$$^{2}$$ - 16y + 64 = 64

⇒ x$$^{2}$$ + y$$^{2}$$ - 16x - 16y + 64 = 0

The Circle