Question

given a circle with the equation $x^{2}+y^{2}-4x + 10y+4 = 0$, what are the coordinates of its center and the length of its radius? use the keypad to enter your answers in the boxes. center ( , ) radius

Explanation:

Step1: Rewrite the circle 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 $x^{2}+y^{2}-4x + 10y+4 = 0$. Complete the square for $x$ and $y$ terms. For the $x$ - terms: $x^{2}-4x=(x - 2)^{2}-4$. For the $y$ - terms: $y^{2}+10y=(y + 5)^{2}-25$. So the equation becomes $(x - 2)^{2}-4+(y + 5)^{2}-25 + 4=0$.

Step2: Simplify the equation.

$(x - 2)^{2}+(y + 5)^{2}=25$.

Step3: Identify the center and radius.

Comparing with the standard - form $(x - a)^2+(y - b)^2=r^2$, we have $a = 2$, $b=-5$ and $r = 5$.

Answer:

Center: $(2,-5)$
Radius: $5$