Question

The equation $x^{2}+y^{2}+4 x+6 y+13=0$ represents
(A) a circle with a radius of 5 units
(B) a pair of two distinct straight lines
(c) a pair of coincident straight lines
(D) a point circle

Answer 1

Hint: A point circle has a radius of 0 and consists entirely of the circle's center. Whether a circle's radius is real or imaginary is determined by the concept of real/imaginary circles. There are no actual points on a circle if the radius is fictitious.

Formula Used:
The radius of the circle is
$r=\sqrt{g^{2}+f^{2}-c}$

Complete step by step Solution:
Given
$x^{2}+y^{2}+4 x+6 y+13=0$
The equation for a circle has the generic form: ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$. The coordinates of the circle's center and radius are found using this general form, where g, f, and c are constants. The general form of the equation of a circle makes it difficult to identify any significant properties about any specific circle, in contrast to the standard form, which is simpler to comprehend. So, to quickly change from the generic form to the standard form, we will use the completing square formula.
Here, $g=2, f=3, c=13$
The radius of the circle is
$r=\sqrt{g^{2}+f^{2}-c}$
$\Rightarrow r=\sqrt{2^{2}+3^{2}-13}$
$\Rightarrow r=0$
$\therefore$ The given equation represents a point circle.

Hence, the correct option is (D).

Note: The radius of a circle is the length of the straight line that connects the center to any point on its circumference. Because a circle's circumference can contain an endless number of points, a circle can have more than one radius. This indicates that a circle has an endless number of radii and that each radius is equally spaced from the circle's center. When the radius's length varies, the circle's size also changes.