General Form of the Equation of a Circle

We will discuss about the general form of the equation of a circle.

Prove that the equation x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 always represents a circle whose centre is (-g, -f) and radius = $\sqrt{g^{2} + f^{2} - c}$, where g, f and c are three constants

 Conversely, a quadratic equation in x and y of the form x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 always represents the equation of a circle.

We know that the equation of the circle having centre at (h, k) and radius = r units is

(x - h)$^{2}$ + (y - k)$^{2}$ = r$^{2}$

⇒ x$^{2}$ + y$^{2}$ - 2hx - 2hy + h$^{2}$ + k$^{2}$ = r$^{2}$

⇒ x$^{2}$ + y$^{2}$ - 2hx - 2hy + h$^{2}$ + k$^{2}$ - r$^{2}$ = 0

Compare the above equation x$^{2}$ + y$^{2}$ - 2hx - 2hy + h$^{2}$ + k$^{2}$ - r$^{2}$ = 0 with x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 we get, h = -g, k = -f and h$^{2}$ + k$^{2}$ - r$^{2}$ = c

Therefore the equation of any circle can be expressed in the form x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0.

Again, x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0

⇒ (x$^{2}$ + 2gx + g$^{2}$) + (y$^{2}$ + 2fy + f$^{2}$) = g$^{2}$ + f$^{2}$ - c

⇒ (x + g)$^{2}$ + (y + f)$^{2}$ = $(\sqrt{g^{2} + f^{2} - c})^{2}$

⇒ {x - (-g) }$^{2}$ + {y - (-f) }$^{2}$ = $(\sqrt{g^{2} + f^{2} - c})^{2}$

This is of the form (x - h)$^{2}$ + (y - k)$^{2}$ = r$^{2}$ which represents a circle having centre at (- g, -f) and radius $\sqrt{g^{2} + f^{2} - c}$.

Hence the given equation x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 represents a circle whose centre is (-g, -f) i.e, (-$\frac{1}{2}$ coefficient of x, -$\frac{1}{2}$ coefficient of y) and radius = $\sqrt{g^{2} + f^{2} - c}$ = $\sqrt{(\frac{1}{2}\textrm{coefficient of x})^{2} + (\frac{1}{2}\textrm{coefficient of y})^{2} - \textrm{constant term}}$

Note:

(i) The equation x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 represents a circle of radius = $\sqrt{g^{2} + f^{2} - c}$.

(ii) If g$^{2}$ + f$^{2}$ - c > 0, then the radius of the circle is real and hence the equation x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 represents a real circle.

(iii) If g$^{2}$ + f$^{2}$ - c = 0 then the radius of the circle becomes zero. In this case, the circle reduces to the point (-g, -f). Such a circle is known as a point circle. In other words, the equation x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 represents a point circle.

(iv) If g$^{2}$ + f$^{2}$ - c < 0, the radius of the circle $\sqrt{g^{2} + f^{2} - c}$ becomes imaginary but the circle is real. Such a circle is called an imaginary circle. In other words, equation x$^{2}$ + y$^{2}$ + 2gx + 2fy + c = 0 does not represent any real circle as it is not possible to draw such a circle.

● 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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