Question

A circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ passing through$(4,-2)$ is concentric to the circle $x^{2}+y^{2}-2 x+4 y+20=0$, then the value of $c$ will be
A. -4
B. 0
C. 4
D. 1

Answer 1

Hint: The center of a circle is a location inside the circle that is situated in the middle of the circumference.The radius of a circle is the constant distance from the circle's center to any point on the circle.A circle's diameter is defined as the segment of a line that connects two locations on the circle and passes through its center.

Complete step by step solution:
Since the circle $S_{1} \equiv 3 x^{2}+y^{2}+2 g x+2 f y+c=0$ is concentric to $S_{2} \equiv x^{2}+y^{2}-$ $2 x+4 y+20=0$
$\therefore$ both have the same center.
center of $\mathrm{S}_{2}=\left(\dfrac{-1}{2} \times-2, \dfrac{1}{2} \times 4\right)=(-1,2)$
$\therefore$ for $\mathrm{S}_{1} g=-1 \mathrm{f}=2$
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 the square formula.
$\therefore$ Equation of circle $\mathrm{S}_{1}$ is $\mathrm{x}^{2}+\mathrm{y}^{2}-2 \mathrm{x}+4 \mathrm{y}+\mathrm{c}=0$ as it passes through $(4,-2)$
$\therefore x^{2}+y^{2}-2 x+4 y+c=0$
$\Rightarrow 4^{2}+(-2)^{2}-2 \times 4+4 \times-2+c=0$
$\Rightarrow 16+4-8-8+c=0$
$\Rightarrow \mathrm{c}=-4$

Option ‘C’ is correct

Note: If two objects in geometry have a common center, they are said to be concentric. Due to their shared center, regular polygons, regular polyhedra, and circles are all concentric. Two circles that are concentric in Euclidean geometry always have different radii but the same center.
Circles having the same or a shared center are known as concentrators. In other terms, circles are said to be concentric if two or more of them share the same center.