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
● The Circle
11 and 12 Grade Math
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