Equations of Concentric Circles

We will learn how to form the equation of concentric circles.

Two circles or more than that are said to be concentric if they have the same centre but different radii.

Let, x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 be a given circle having centre at (- g, - f) and radius = \(\mathrm{\sqrt{g^{2} + f^{2} - c}}\).

Therefore, the equation of a circle concentric with the given circle x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c = 0 is

x\(^{2}\) + y\(^{2}\) + 2gx + 2fy + c' = 0 

Both the circle have the same centre (- g, - f) but their radii are not equal (since, c ≠ c')

Similarly, the equation of a circle with centre at (h, k) and radius equal to r, is (x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(^{2}\).

Therefore, the equation of a circle concentric with the circle (x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(^{2}\) is  (x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(_{1}\)\(^{2}\), (r\(_{1}\) ≠ r)      

Assigning different values to r\(_{1}\) we shall have a family of circles each of which is concentric with the circle (x - h)\(^{2}\) + (y - k)\(^{2}\) = r\(^{2}\).

Solved example to find the equation of a concentric circle:

Find the equation of the circle which is concentric with the circle 2x\(^{2}\) + 2y\(^{2}\) + 3x - 4y + 5 = 0 and whose radius is 2√5 units.

Solution:         

2x\(^{2}\) + 2y\(^{2}\) + 3x - 4y + 5 = 0

⇒ x\(^{2}\) + y\(^{2}\) + 3/2x - 2y + \(\frac{5}{2}\) = 0 ………………..(i)

Clearly, the equation of a circle concentric with the circle (i) is

x\(^{2}\) + y\(^{2}\) + \(\frac{3}{2}\)x - 2y + c = 0 ……………………..(ii)

Now, the radius of the circle (ii) = \(\sqrt{(\frac{3}{2})^{2} + (-2)^{2} - c}\)

By question, \(\sqrt{\frac{9}{4} + 4 - c}\) = 2√5

⇒ \(\frac{25}{4}\) - c = 20

⇒ c = \(\frac{25}{4}\) - 20

c = -\(\frac{55}{4}\)

Therefore, the equation of the required circle is

x\(^{2}\) + y\(^{2}\) + \(\frac{3}{2}\)x - 2y - \(\frac{55}{4}\) = 0

⇒ 4x\(^{2}\) + 4y\(^{2}\) + 6x - 8y - 55 = 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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