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
11 and 12 Grade Math
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