Question

The equation ${x^2} + {y^2} + 2gx + 2fy + c = 0$ always represent a circle whose centre is $( - g, - f)$ and radius is $\sqrt {{g^2} + {f^2} - c} $. If ${g^2} + {f^2} = c$, then in this case, the circle is called as:
A Ordinary circle
B Incircle
C Degenerate circle
D None of these

Answer 1

Hint: First we will solve the given equation ${x^2} + {y^2} + 2gx + 2fy + c = 0$ to transform that in the form of equation having centre is $( - g, - f)$ and radius is $\sqrt {{g^2} + {f^2} - c} $. Then we will use the given condition ${g^2} + {f^2} = c$ to identify the type of circle.

Complete step by step Solution:
${x^2} + {y^2} + 2gx + 2fy + c = 0$
Adding and subtracting ${g^2}$ and ${f^2}$
${x^2} + 2gx + {g^2} + {y^2} + 2fy + {f^2} + c - {g^2} - {f^2} = 0$
After simplifying the above equation
${(x + g)^2} + {(y + f)^2} + c - {g^2} - {f^2} = 0$
Shifting ${g^2},{f^2},c$ to the other side
${(x + g)^2} + {(y + f)^2} = {g^2} + {f^2} - c$
Transforming the above equation into the form of an equation of a circle
${(x + g)^2} + {(y + f)^2} = {\left( {\sqrt {{g^2} + {f^2} - c} } \right)^2}$ (1)
Comparing with the equation of circle ${(x - h)^2} + {\left( {y - k} \right)^2} + {r^2}$ having center $(h,k)$ and radius $r$
Therefore, the center of given circle is $( - g, - f)$ and radius is $\sqrt {{g^2} + {f^2} - c} $
If ${g^2} + {f^2} = c$
Putting in the equation (1)
${(x + g)^2} + {(y + f)^2} = {\left( {\sqrt {c - c} } \right)^2}$
After solving the above equation
${(x + g)^2} + {(y + f)^2} = 0$
As we can see that the radius vanishes.
Hence, the circle becomes the point. And such circles are called point circles or degenerated circles.

Therefore, the correct option is C.

Additional Information: A degenerate case in mathematics is a limiting case of a class of objects that seems to be qualitatively different from (and typically simpler than) the remainder of the class. The condition of being a degenerate case is called degeneracy.
Many types of composite or structured objects have implicit inequalities in their definitions. For instance, a triangle should have positive angles and side lengths. Degeneracies are the limiting circumstances in which one or more of these inequalities turn into equalities. A triangle is considered to be degenerate if at least one of its side lengths or angles is zero. In this case, it is equivalent to a "line segment."

Note:Students should concentrate on the concept that they have been given and should proceed according to that to avoid any mistakes. They can make mistakes while transforming the given equation of a circle to the standard equation of a circle.