# Centre of the Circle on x-axis

We will learn how to find the equation when the centre of a circle on 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 centre of a circle is on the x-axis i.e., k = 0.

Then the equation (x - h)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ becomes (x - h)$$^{2}$$ + y$$^{2}$$ = a$$^{2}$$ ⇒ x$$^{2}$$ + y$$^{2}$$ - 2hx + h$$^{2}$$ = a$$^{2}$$ ⇒ x$$^{2}$$ + y$$^{2}$$ - 2hx + h$$^{2}$$ – a$$^{2}$$ = 0

If the centre of a circle be on the x-axis, then the y co-ordinate of the centre will be zero. Hence, the general form of the equation of the circle will be of the form x$$^{2}$$ + y$$^{2}$$ + 2gx + c = 0, where g and c are the constants.

Solved examples on the central form of the equation of a circle whose centre is on the x-axis:

1. Find the equation of a circle whose centre of a circle is on the x-axis at -5 and radius is 9 units.

Solution:

Radius of the circle = 9 units.

Since, centre of a circle be on the x-axis, then the y co-ordinate of the centre will be zero.

The required equation of the circle whose centre of a circle is on the x-axis at -5 and radius is 9 units is

(x + 5)$$^{2}$$ + y$$^{2}$$ = 9$$^{2}$$

⇒ x$$^{2}$$ + 10x + 25 + y$$^{2}$$ = 81

⇒ x$$^{2}$$ + y$$^{2}$$ + 10x  + 25 - 81 = 0

⇒ x$$^{2}$$ + y$$^{2}$$ + 10x  - 56 = 0

2. Find the equation of a circle whose centre of a circle is on the x-axis at 2 and radius is 3 units.

Solution:

Radius of the circle = 3 units.

Since, centre of a circle be on the x-axis, then the y co-ordinate of the centre will be zero.

The required equation of the circle whose centre of a circle is on the x-axis at 2 and radius is 3 units is

(x - 2)$$^{2}$$ + y$$^{2}$$ = 3$$^{2}$$

⇒ x$$^{2}$$ - 4x + 4 + y$$^{2}$$ = 9

⇒ x$$^{2}$$ + y$$^{2}$$ - 4x  + 4 - 9 = 0

⇒ x$$^{2}$$ + y$$^{2}$$ - 4x  - 5 = 0