Centre of the Circle on y-axis

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

• Definition of Circle
• Equation of a Circle
• General Form of the Equation of a Circle
• General Equation of Second Degree Represents a Circle
• Centre of the Circle Coincides with the Origin
• Circle Passes through the Origin
• Circle Touches x-axis
• Circle Touches y-axis
• Circle Touches both x-axis and y-axis
• Centre of the Circle on x-axis
• Centre of the Circle on y-axis
• Circle Passes through the Origin and Centre Lies on x-axis
• Circle Passes through the Origin and Centre Lies on y-axis
• Equation of a Circle when Line Segment Joining Two Given Points is a Diameter
• Equations of Concentric Circles
  
• Circle Passing Through Three Given Points
• Circle Through the Intersection of Two Circles
• Equation of the Common Chord of Two Circles
• Position of a Point with Respect to a Circle
• Intercepts on the Axes made by a Circle
• Circle Formulae
• Problems on Circle

11 and 12 Grade Math 

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