# Circle Touches x-axis

We will learn how to find the equation of a circle touches x-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 x-axis i.e., k = a.

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

If a circle touches the x-axis, then the y-co-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 - h)$$^{2}$$ + (y - a)$$^{2}$$ = a$$^{2}$$

Let C (h, k) be the centre of the circle. Since the circle touches the x-axis, therefore, a = k

Hence the equation of the circle is (x - h)$$^{2}$$ + (y - a)$$^{2}$$ = a$$^{2}$$ ⇒ x$$^{2}$$ + y$$^{2}$$ - 2hx - 2ay + h$$^{2}$$ = 0

Solved examples on the central form of the equation of a circle touches x-axis:

1. Find the equation of a circle whose x-coordinate of the centre is 5 and radius is 4 units also touches the x-axis.

Solution:

The required equation of the circle whose x-coordinate of the centre is 5 and radius is 4 units also touches the x-axis is (x - 5)$$^{2}$$ + (y - 4)$$^{2}$$ = 4$$^{2}$$, [Since radius is equal to the y-coordinate of the centre]

⇒ x$$^{2}$$ – 10x + 25 + y$$^{2}$$ – 8y + 16 = 16

⇒ x$$^{2}$$ + y$$^{2}$$ - 10x - 8y + 25 = 0

2. Find the equation of a circle whose radius is 7 units and x-coordinate of the centre is -2 and also touches the x-axis.

Solution:

The required equation of the circle whose radius is 7 units and x-coordinate of the centre is -2 and also touches the x-axis is (x + 2)$$^{2}$$ + (y - 7)$$^{2}$$ = 7$$^{2}$$, [Since radius is equal to the y-coordinate of the centre]

⇒ x$$^{2}$$ + 4x + 4 + y$$^{2}$$ – 14y + 49 = 49

⇒ x$$^{2}$$ + y$$^{2}$$ + 4x - 14y + 4 = 0