The equation x2 + y2 + 2gx + 2fy + c = 0 always represents a circle whose centre is (-g, -f) and radius is \(\sqrt {{g^2} + {f^2} - c} \). If g2 + f2 = c, then in this case, the circle is called as:
degenerate circle
Let's analyze the given equation of a circle and the condition provided.
The general equation of a circle is given by:
\(x^2 + y^2 + 2gx + 2fy + c = 0\)
From this standard form, we know the properties of the circle:
- The centre of the circle is at the point \((-g, -f)\).
- The radius of the circle is given by the formula \(r = \sqrt{{g^2} + {f^2} - c}\).
The question gives us a specific condition regarding the coefficients g, f, and c:
\(g^2 + f^2 = c\)
Understanding the Radius under the Condition
Let's substitute the given condition, \(g^2 + f^2 = c\), into the formula for the radius, \(r\).
The radius is:
\(r = \sqrt{{g^2} + {f^2} - c}\)
Substitute \(c\) with \((g^2 + f^2)\):
\(r = \sqrt{(g^2 + f^2) - (g^2 + f^2)}\)
\(r = \sqrt{g^2 + f^2 - g^2 - f^2}\)
\(r = \sqrt{0}\)
\(r = 0\)
So, under the condition \(g^2 + f^2 = c\), the radius of the circle is 0.
What is a Circle with Zero Radius?
A circle is typically defined as the set of all points in a plane that are a fixed distance (the radius) from a fixed point (the centre). If the radius is 0, the only point that satisfies this condition is the centre itself. Therefore, a "circle" with a radius of 0 is just a single point.
In geometry, when an equation that usually represents a curve (like a circle) represents a simpler form like a point or a line under certain conditions, it is called a 'degenerate' case of that curve.
For a circle, the degenerate case occurs when the radius is zero. This happens when \(g^2 + f^2 - c = 0\), or \(g^2 + f^2 = c\).
Such a circle is referred to as a degenerate circle.
Analyzing the Options
Let's look at the given options:
- ordinary circle: An ordinary circle has a positive radius (\(r > 0\)), meaning \(g^2 + f^2 - c > 0\). This is not the case here.
- incircle: An incircle is a circle inscribed within a triangle, tangent to all three sides. This term is specific to a triangle's geometry, not the general condition \(g^2 + f^2 = c\).
- degenerate circle: As discussed, when the radius of a circle is 0 (\(g^2 + f^2 - c = 0\)), the circle reduces to a single point, which is called a degenerate circle. This matches our finding.
- none of these: This is incorrect because 'degenerate circle' is a valid description for this case.
Therefore, if \(g^2 + f^2 = c\), the circle is called a degenerate circle.
| Condition on Radius (\(r = \sqrt{{g^2} + {f^2} - c}\)) | Condition on \(g^2 + f^2 - c\) | Type of Circle |
|---|---|---|
| \(r > 0\) | \(g^2 + f^2 - c > 0\) | Real Circle (Ordinary Circle) |
| \(r = 0\) | \(g^2 + f^2 - c = 0\) | Degenerate Circle (A Point) |
| \(r\) is imaginary | \(g^2 + f^2 - c < 0\) | Imaginary Circle (Does not exist in the real plane) |
Revision Table: Circle Equation and Conditions
| Equation | Centre | Radius (\(r\)) | Condition for Real Circle | Condition for Degenerate Circle | Condition for Imaginary Circle |
|---|---|---|---|---|---|
| \(x^2 + y^2 + 2gx + 2fy + c = 0\) | \((-g, -f)\) | \(\sqrt{{g^2} + {f^2} - c}\) | \(g^2 + f^2 - c > 0\) | \(g^2 + f^2 - c = 0\) | \(g^2 + f^2 - c < 0\) |
Additional Information on Degenerate Conics
The circle is a type of conic section. Conic sections (like circles, ellipses, parabolas, and hyperbolas) are formed by the intersection of a plane and a double cone. Degenerate conic sections occur when the plane passes through the vertex of the cone.
- A degenerate circle is a point.
- A degenerate ellipse is a point.
- A degenerate parabola is a line (or two parallel lines).
- A degenerate hyperbola is a pair of intersecting lines.
In the case of the circle equation \(x^2 + y^2 + 2gx + 2fy + c = 0\), the degenerate case \(g^2 + f^2 - c = 0\) results in the equation \((x+g)^2 + (y+f)^2 = 0\). The only real solution for this equation is when \(x+g=0\) and \(y+f=0\), which means \(x=-g\) and \(y=-f\). This is a single point \((-g, -f)\), which is the centre of the degenerate circle.
Important Questions from Equation of Circle
The equation of circle with centre (1, -2) and radius 4 cm is:
The intercept on the line y = x by the circle x 2+ y 2- 2x = 0 is AB. Equation of circle with AB as diameter is
If the equation x 2+ y 2 - 4x - 4y + 4 = 0 represents a circle, then its radius is
Radius of the circle x 2+ y 2– 4x + 2y – 31 = 0 is
Find the equation of the circle which passes through (-1, 1) and (2, 1), and having centre on the line x + 2y + 3 = 0.
