Question
A circle is represented by the equation x2+y2−16x−6y=−24
. Complete the square to discover the center and radius of the circle.(1 point)
Responses
center=(−16,−6)
, radius=8
center= left parenthesis negative 16 comma negative 6 right parenthesis , radius=8
center=(16,6)
, radius=8
center= left parenthesis 16 comma 6 right parenthesis , radius=8
center=(8,3)
, radius=7
center= left parenthesis 8 comma 3 right parenthesis , radius=7
center=(3,8)
, radius=7
. Complete the square to discover the center and radius of the circle.(1 point)
Responses
center=(−16,−6)
, radius=8
center= left parenthesis negative 16 comma negative 6 right parenthesis , radius=8
center=(16,6)
, radius=8
center= left parenthesis 16 comma 6 right parenthesis , radius=8
center=(8,3)
, radius=7
center= left parenthesis 8 comma 3 right parenthesis , radius=7
center=(3,8)
, radius=7
Answers
Answered by
GPT-4o mini
To find the center and radius of the circle given by the equation \( x^2 + y^2 - 16x - 6y = -24 \), we need to complete the square for both the \( x \) and \( y \) terms.
Starting with the equation:
\[
x^2 + y^2 - 16x - 6y = -24
\]
### Step 1: Rearranging
We'll rearrange the equation into a form that allows us to complete the square:
\[
x^2 - 16x + y^2 - 6y = -24
\]
### Step 2: Completing the square for \( x \)
Take the \( x \) terms:
\[
x^2 - 16x
\]
To complete the square, take half of \(-16\) (which is \(-8\)), square it to get \(64\), and rewrite:
\[
x^2 - 16x + 64 - 64
\]
This simplifies to:
\[
(x - 8)^2 - 64
\]
### Step 3: Completing the square for \( y \)
Now take the \( y \) terms:
\[
y^2 - 6y
\]
Take half of \(-6\) (which is \(-3\)), square it to get \(9\), and rewrite:
\[
y^2 - 6y + 9 - 9
\]
This simplifies to:
\[
(y - 3)^2 - 9
\]
### Step 4: Putting it all together
Now substitute back into the equation:
\[
(x - 8)^2 - 64 + (y - 3)^2 - 9 = -24
\]
Combine constants on the left-hand side:
\[
(x - 8)^2 + (y - 3)^2 - 73 = -24
\]
Bring \(-73\) to the right side:
\[
(x - 8)^2 + (y - 3)^2 = -24 + 73
\]
This simplifies to:
\[
(x - 8)^2 + (y - 3)^2 = 49
\]
### Step 5: Determine center and radius
From the final equation \((x - 8)^2 + (y - 3)^2 = 49\):
- The center of the circle is \((8, 3)\).
- The radius is \(\sqrt{49} = 7\).
Thus, the center and radius of the circle are:
center = (8, 3), radius = 7
So the correct response is:
**center=(8,3), radius=7**
Starting with the equation:
\[
x^2 + y^2 - 16x - 6y = -24
\]
### Step 1: Rearranging
We'll rearrange the equation into a form that allows us to complete the square:
\[
x^2 - 16x + y^2 - 6y = -24
\]
### Step 2: Completing the square for \( x \)
Take the \( x \) terms:
\[
x^2 - 16x
\]
To complete the square, take half of \(-16\) (which is \(-8\)), square it to get \(64\), and rewrite:
\[
x^2 - 16x + 64 - 64
\]
This simplifies to:
\[
(x - 8)^2 - 64
\]
### Step 3: Completing the square for \( y \)
Now take the \( y \) terms:
\[
y^2 - 6y
\]
Take half of \(-6\) (which is \(-3\)), square it to get \(9\), and rewrite:
\[
y^2 - 6y + 9 - 9
\]
This simplifies to:
\[
(y - 3)^2 - 9
\]
### Step 4: Putting it all together
Now substitute back into the equation:
\[
(x - 8)^2 - 64 + (y - 3)^2 - 9 = -24
\]
Combine constants on the left-hand side:
\[
(x - 8)^2 + (y - 3)^2 - 73 = -24
\]
Bring \(-73\) to the right side:
\[
(x - 8)^2 + (y - 3)^2 = -24 + 73
\]
This simplifies to:
\[
(x - 8)^2 + (y - 3)^2 = 49
\]
### Step 5: Determine center and radius
From the final equation \((x - 8)^2 + (y - 3)^2 = 49\):
- The center of the circle is \((8, 3)\).
- The radius is \(\sqrt{49} = 7\).
Thus, the center and radius of the circle are:
center = (8, 3), radius = 7
So the correct response is:
**center=(8,3), radius=7**