Question

question 8 of 30 which conic section is defined by the equation shown below? $x^{2}+y^{2}-10x - 2y+10 = 0$ a. ellipse b. hyperbola c. circle d. parabola

Explanation:

Step1: Complete the square for x and y terms.

For the x - terms:
\[

$$\begin{align*} x^{2}-10x&=(x - 5)^{2}-25 \end{align*}$$

\]
For the y - terms:
\[

$$\begin{align*} y^{2}-2y&=(y - 1)^{2}-1 \end{align*}$$

\]
The original equation \(x^{2}+y^{2}-10x - 2y + 10=0\) can be rewritten as \((x - 5)^{2}-25+(y - 1)^{2}-1 + 10=0\).

Step2: Simplify the equation.

\[

$$\begin{align*} (x - 5)^{2}+(y - 1)^{2}-25-1 + 10&=0\\ (x - 5)^{2}+(y - 1)^{2}&=16 \end{align*}$$

\]
The general equation of a circle is \((x - a)^{2}+(y - b)^{2}=r^{2}\), where \((a,b)\) is the center of the circle and \(r\) is the radius.

Answer:

C. Circle