# Circle Touches y-axis

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

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

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

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

Hence the equation of the circle is (x - a)$$^{2}$$ + (y - k)$$^{2}$$ = a$$^{2}$$ ⇒ x$$^{2}$$ + y$$^{2}$$ – 2ax – 2ky + k$$^{2}$$ = 0

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

1. Find the equation of a circle whose y-coordinate of the centre is -7 and radius is 3 units also touches the y-axis.

Solution:

The required equation of the circle whose y-coordinate of the centre is -7 and radius is 3 units also touches the y-axis is (x - 3)$$^{2}$$ + (y + 7)$$^{2}$$ = 3$$^{2}$$, [Since radius is equal to the x-coordinate of the centre]

⇒ x$$^{2}$$ – 6x + 9 + y$$^{2}$$ + 14y + 49 = 9

⇒ x$$^{2}$$ + y$$^{2}$$ - 6x  + 14y + 49 = 0

2. Find the equation of a circle whose radius is 9 units and y-coordinate of the centre is -6 and also touches the y-axis.

Solution:

The required equation of the circle whose radius is 9 units and y-coordinate of the centre is -6 and also touches the x-axis is (x - 9)$$^{2}$$ + (y + 6)$$^{2}$$ = 9$$^{2}$$, [Since radius is equal to the x-coordinate of the centre]

⇒ x$$^{2}$$ - 18x + 81 + y$$^{2}$$ + 12y + 36 = 81

⇒ x$$^{2}$$ + y$$^{2}$$ - 18x + 12y + 36 = 0

The Circle