We will learn how to find the equation when the centre of a circle on 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 centre of a circle is on the y-axis i.e., h = 0.
Then the equation (x - h)\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\) becomes x\(^{2}\) + (y - k)\(^{2}\) = a\(^{2}\) ⇒ x\(^{2}\) + y\(^{2}\) - 2ky + k\(^{2}\) = a\(^{2}\) ⇒ x\(^{2}\) + y\(^{2}\) - 2ky + k\(^{2}\) - a\(^{2}\) = 0
If the centre of a circle be on the y-axis, then the x co-ordinate of the centre will be zero. Hence, the general form of the equation of the circle will be of the form x2 + y2 + 2fy + c = 0, where g and c are the constants.
Solved examples on
the central form of the equation of a circle whose centre is on the y-axis:
1. Find the equation of a circle whose centre of a circle is on the y-axis at -3 and radius is 6 units.
Solution:
Radius of the circle = 6 units.
Since, centre of a circle be on the y-axis, then the x co-ordinate of the centre will be zero.
The required equation of the circle whose centre of a circle is on the y-axis at -3 and radius is 6 units is
x\(^{2}\) + (y + 3)\(^{2}\) = 6\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) + 6y + 9 = 36
⇒ x\(^{2}\) + y\(^{2}\) + 6y + 9 - 36 = 0
⇒ x\(^{2}\) + y\(^{2}\) + 6y - 27 = 0
2. Find the equation of a circle whose centre of a circle is on the y-axis at 4 and radius is 4 units.
Solution:
Radius of the circle = 4 units.
Since, centre of a circle be on the y-axis, then the x co-ordinate of the centre will be zero.
The required equation of the circle whose centre of a circle is on the y-axis at 4 and radius is 4 units is
x\(^{2}\) + (y - 4)\(^{2}\) = 4\(^{2}\)
⇒ x\(^{2}\) + y\(^{2}\) - 8y + 16 = 16
⇒ x\(^{2}\) + y\(^{2}\) – 8y + 16 - 16 = 0
⇒ x\(^{2}\) + y\(^{2}\) - 8y = 0
● The Circle
11 and 12 Grade Math
From Centre of the Circle on y-axis to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Apr 22, 24 05:19 PM
Apr 22, 24 01:35 PM
Apr 21, 24 10:57 AM
Apr 20, 24 05:39 PM
Apr 20, 24 05:29 PM