Question

Math

Posted 2 months ago

```
A certain circle can be represented by the following equation.
$x^{2}+y^{2}-20 x+8 y+115=0$
What is the center of this circle?
$\square$ , $\square$
What is the radius of this circle?
$\square$ units
```

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

To find the center of the circle, we need to complete the square for the $x$ and $y$ terms in the equation

step 2

The given equation is $x^2 + y^2 - 20x + 8y + 115 = 0$. We group the $x$ terms and the $y$ terms: $(x^2 - 20x) + (y^2 + 8y) + 115 = 0$

step 3

To complete the square for the $x$ terms, we take half of the coefficient of $x$, which is $-20$, and square it, giving $(\frac{-20}{2})^2 = 100$. We add and subtract $100$ inside the equation

step 4

To complete the square for the $y$ terms, we take half of the coefficient of $y$, which is $8$, and square it, giving $(\frac{8}{2})^2 = 16$. We add and subtract $16$ inside the equation

step 5

The equation now looks like this: $(x^2 - 20x + 100) + (y^2 + 8y + 16) + 115 - 100 - 16 = 0$

step 6

We can rewrite the equation as $(x - 10)^2 + (y + 4)^2 - 1 = 0$

step 7

Rearranging the equation, we get $(x - 10)^2 + (y + 4)^2 = 1$

step 8

The center of the circle is at the point $(h, k)$ where $h$ and $k$ are the values that complete the square, so the center is $(10, -4)$

step 9

The radius of the circle is the square root of the constant on the right side of the equation, which is $\sqrt{1} = 1$

[question 1] Answer

The center of the circle is $(10, -4)$.

The radius of the circle is $1$ unit.

Key Concept

Completing the square to find the center and radius of a circle

Explanation

The center of a circle in the equation $(x - h)^2 + (y - k)^2 = r^2$ is $(h, k)$, and the radius is $r$. By completing the square, we can rewrite the given equation in this form to find the center and radius.

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