We will learn how to find the equation of a circle touches xaxis.
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 xaxis 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 xaxis, then the ycoordinate 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 xaxis, 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 xaxis:
1. Find the equation of a circle whose xcoordinate of the centre is 5 and radius is 4 units also touches the xaxis.
Solution:
The required equation of the circle whose xcoordinate of the centre is 5 and radius is 4 units also touches the xaxis is (x  5)\(^{2}\) + (y  4)\(^{2}\) = 4\(^{2}\), [Since radius is equal to the ycoordinate 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 xcoordinate of the centre is 2 and also touches the xaxis.
Solution:
The required equation of the circle whose radius is 7 units and xcoordinate of the centre is 2 and also touches the xaxis is (x + 2)\(^{2}\) + (y  7)\(^{2}\) = 7\(^{2}\), [Since radius is equal to the ycoordinate of the centre]
⇒ x\(^{2}\) + 4x + 4 + y\(^{2}\) – 14y + 49 = 49
⇒ x\(^{2}\) + y\(^{2}\) + 4x  14y + 4 = 0
11 and 12 Grade Math
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