Martin Conner
07/14/2023 · Elementary School
\( (x+4)^{2}+(y+3)^{2}=49 \) Select the correct choice below and, if necessary, fill in the answer box within your choice. A. The x-intercept(s) is/are \( 2 \sqrt{10}-4,-2 \sqrt{10}-4 \). There is no \( x \)-intercept. (Use a comma to separate answers as needed. Type an exact answer, using radicals as needed.) A. The \( y \)-intercept(s) is/are (Use a comma to separate answers as needed. Type an exact answer, using radicals as needed.) B. There is no \( y \)-intercept.
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Tutor-Verified Answer
Step-by-step Solution
The given equation is
\[
(x+4)^{2} + (y+3)^{2} = 49
\]
This represents a circle with center at \((-4, -3)\) and a radius of \(7\) (since \( \sqrt{49} = 7\)).
### Finding the x-intercepts:
To find the x-intercepts, we set \(y = 0\) and solve for \(x\):
\[
(x+4)^{2} + (0+3)^{2} = 49
\]
This simplifies to:
\[
(x+4)^{2} + 9 = 49
\]
Subtracting \(9\) from both sides gives:
\[
(x+4)^{2} = 40
\]
Taking the square root of both sides:
\[
x + 4 = \pm \sqrt{40}
\]
Since \(\sqrt{40} = 2\sqrt{10}\), we have:
\[
x + 4 = 2\sqrt{10} \quad \text{or} \quad x + 4 = -2\sqrt{10}
\]
Solving for \(x\):
1. \(x = 2\sqrt{10} - 4\)
2. \(x = -2\sqrt{10} - 4\)
Thus, the x-intercepts are:
\[
2\sqrt{10} - 4, -2\sqrt{10} - 4
\]
### Finding the y-intercepts:
To find the y-intercepts, we set \(x = 0\) and solve for \(y\):
\[
(0+4)^{2} + (y+3)^{2} = 49
\]
This simplifies to:
\[
16 + (y+3)^{2} = 49
\]
Subtracting \(16\) from both sides gives:
\[
(y+3)^{2} = 33
\]
Taking the square root of both sides:
\[
y + 3 = \pm \sqrt{33}
\]
Solving for \(y\):
1. \(y = \sqrt{33} - 3\)
2. \(y = -\sqrt{33} - 3\)
Thus, the y-intercepts are:
\[
\sqrt{33} - 3, -\sqrt{33} - 3
\]
### Final Answers:
A. The x-intercept(s) is/are \( 2\sqrt{10} - 4, -2\sqrt{10} - 4 \).
A. The y-intercept(s) is/are \( \sqrt{33} - 3, -\sqrt{33} - 3 \).
Quick Answer
A. The x-intercept(s) is/are \( 2\sqrt{10} - 4, -2\sqrt{10} - 4 \).
A. The y-intercept(s) is/are \( \sqrt{33} - 3, -\sqrt{33} - 3 \).
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