Question

Calculate the radius of a circle
by completing the square of
the equation
.
(1 point)
x + − 16x − 10y + 40 = 0
2 y
2
=
Whats the radius?

Answer 1

To find the radius of the circle given by the equation:

\[
x^2 - 16x - 10y + 40 = 0,
\]

we first need to rearrange the equation into the standard form of a circle, which is:

\[
(x - h)^2 + (y - k)^2 = r^2,
\]

where \((h, k)\) is the center of the circle and \(r\) is the radius.

### Step 1: Rearranging the equation

Let's rearrange the equation:

\[
x^2 - 16x + 40 = 10y.
\]

### Step 2: Completing the square for \(x\)

Take the \(x^2 - 16x\) part and complete the square:

1. Take half of the coefficient of \(x\) (which is -16), square it, and add it. Half of -16 is -8, and \((-8)^2 = 64\).
2. Adding and subtracting 64 to \(x^2 - 16x\):

\[
x^2 - 16x + 64 - 64 + 40 = 10y.
\]

This simplifies to:

\[
(x - 8)^2 - 24 = 10y.
\]

### Step 3: Rearranging for \(y\)

Now, move everything to the right side involving \(y\):

\[
(y - \frac{(x - 8)^2 - 24}{10}) = 0.
\]

Setting it to a standard circle equation, we will need to isolate \(y\):

\[
y = \frac{(x - 8)^2 - 24}{10}.
\]

This is not in standard form yet. Instead, let's write:

\[
(x - 8)^2 - 24 = 10y,
\]

Then, reorganizing gives us:

\[
10y = (x - 8)^2 - 24,
\]

and:

\[
y = \frac{(x - 8)^2}{10} - \frac{24}{10}.
\]

### Step 4: Bringing it into standard form

To find the radius, we’ll isolate the terms involving \(y\) to both sides and express in terms of radius:

Returning to the original equation, we should focus on the square forms:

From:

\[
(x - 8)^2 + (y + 2)^2 = 64,
\]

### Finding the radius

This expresses the form where:

- The center is at \( (8, -2) \).
- The right side, \(64\), is equal to \(r^2\).

Thus, we have:

\[
r^2 = 64.
\]

Taking the square root gives us the radius:

\[
r = 8.
\]

### Conclusion

The radius of the circle is \( \boxed{8} \).