We will learn how to find the equation of a circle touches yaxis.
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 yaxis 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 yaxis, then the xcoordinate 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 yaxis, 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 yaxis:
1. Find the equation of a circle whose ycoordinate of the centre is 7 and radius is 3 units also touches the yaxis.
Solution:
The required equation of the circle whose ycoordinate of the centre is 7 and radius is 3 units also touches the yaxis is (x  3)\(^{2}\) + (y + 7)\(^{2}\) = 3\(^{2}\), [Since radius is equal to the xcoordinate 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 ycoordinate of the centre is 6 and also touches the yaxis.
Solution:
The required equation of the circle whose radius is 9 units and ycoordinate of the centre is 6 and also touches the xaxis is (x  9)\(^{2}\) + (y + 6)\(^{2}\) = 9\(^{2}\), [Since radius is equal to the xcoordinate of the centre]
⇒ x\(^{2}\)  18x + 81 + y\(^{2}\) + 12y + 36 = 81
⇒ x\(^{2}\) + y\(^{2}\)  18x + 12y + 36 = 0
`11 and 12 Grade Math
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