Question

The equation of circumcircle of the triangle formed by the lines $y+\sqrt{3} x=6, y-$ $\sqrt{3} x=6$ and $y=0$, is
A) ${{x}^{2}}+{{y}^{2}}+4x=0$
B) $x^{2}+y^{2}-4 x=0$
C) ${{x}^{2}}+{{y}^{2}}-4y=12$
D) $x^{2}+y^{2}+4 y=12$

Answer 1

Hint: The general equation of a circle is another name for the center of a circle formula. If the radius is r, the center's coordinates are $(h,k),$and any point on the circle is$(x, y)$, the center of the circle formula is as follows:
$(x-h)^{2}+(y-k)^{2}=r^{2}$
The equation for the center of the circle is another name for this. In the sections that follow, we'll use this formula to determine a circle's equation or center.

Complete step by step Solution:
Given that
$y+\sqrt{3} x=6, y-\sqrt{3} x=6, y=0$
Intersection of $y+\sqrt{3}=6$ and $y-\sqrt{3} x=6$
$\Rightarrow 6+\sqrt{3}+\sqrt{3}=6$
$\Rightarrow \mathrm{x}=0 \Rightarrow \mathrm{y}=6$
Intersection of $y+\sqrt{3}=6$ and $y=0$
$\Rightarrow \mathrm{x}, \mathrm{y}=0$
Intersection of $y-\sqrt{3} x+6=0$ and $y=0$
$\mathrm{x}=-2 \sqrt{3} ; \mathrm{y}=0$
The increase divided by the run, or the ratio of the rise to the run is known as the line's slope. In the coordinate plane, it describes the slope of the line. Finding the slope between two separate points and calculating the slope of a line are related tasks. In general, we require the values of any two separate coordinates on a line in order to determine its slope.
Slope of $y+\sqrt{3} x=6 \Rightarrow m_{1}=-\sqrt{3}$
$\theta_{1}=-60^{\circ}$
Slope of $\mathrm{y}-\sqrt{3} \mathrm{x}=6 \Rightarrow \mathrm{m}_{2}=\sqrt{3}$
$\theta_{2}=60^{\circ}$
$\theta_{3}=180-(60+60)=60^{\circ}$
$\therefore \mathrm{ABC}$ is an equilateral triangle
length $=4 \sqrt{3}$
Circum center = centroid $=(0,2)$
$\text { Radius }=\dfrac{\text { length }}{\sqrt{2}}=4\left[r \cos \theta=\dfrac{a}{2} \Rightarrow \dfrac{\sqrt{3} r}{2}=\dfrac{a}{2} \Rightarrow r=\dfrac{a}{\sqrt{3}}\right]$
$(x-0)^{2}+(y-2)^{2}=r^{2}$
$x^{2}+y^{2}+4-4 y=16$
$\Rightarrow x^{2}+y^{2}-4 y=12$

Hence, the correct option is (C).

Note:All of these polygons are referred to be cyclic polygons since their circumcircles are all circles that go through their vertices or corners. Only regular polygons, triangles, rectangles, and right-kites have circumcircles, hence not all polygons meet this requirement. This circle's radius is known as the circumradius, which is the radius of the polygon's circumcircle, and its center is known as the circumcenter, which is the origin or center of the circle.