Question

given a circle with the equation $x^{2}+y^{2}+10x - 4y + 25 = 0$, what is the coordinate of its center and the length of its radius? enter your answers in the boxes. the coordinate of the circles center is ( , ). the circles radius is units.

Explanation:

Step1: Rewrite the equation in standard form

The general equation of a circle is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center and $r$ is the radius. Given $x^{2}+y^{2}+10x - 4y+25 = 0$. Complete the square for $x$ and $y$ terms. For the $x$ - terms: $x^{2}+10x=(x + 5)^{2}-25$. For the $y$ - terms: $y^{2}-4y=(y - 2)^{2}-4$. So the equation becomes $(x + 5)^{2}-25+(y - 2)^{2}-4 + 25=0$.

Step2: Simplify the equation

Simplify the above - equation: $(x + 5)^{2}+(y - 2)^{2}=4$.

Step3: Identify the center and radius

Comparing $(x + 5)^{2}+(y - 2)^{2}=4$ with $(x - a)^2+(y - b)^2=r^2$, we have $a=-5$, $b = 2$ and $r = 2$.

Answer:

The coordinate of the circle's center is $(-5,2)$.
The circle's radius is $2$ units.