Cruz Ramirez
12/18/2023 · High School
The equation of a circle is given in general form. Complete parts a through d to follow the step by step process to sketch the graph of the circle. \( x^{2}+y^{2}-14 x+2 y+34=0 \) a) Write the equation of the circle in standard form. The equation is (Simplify your answer.)
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Tutor-Verified Answer
Step-by-step Solution
To convert the given equation of the circle from general form to standard form, we need to complete the square for both the \(x\) and \(y\) terms.
The given equation is:
\[
x^{2} + y^{2} - 14x + 2y + 34 = 0
\]
First, we can rearrange the equation by moving the constant term to the other side:
\[
x^{2} - 14x + y^{2} + 2y = -34
\]
Next, we will complete the square for the \(x\) terms and the \(y\) terms.
### Completing the square for \(x\):
1. Take the coefficient of \(x\) (which is \(-14\)), divide it by 2, and square it:
\[
\left(-\frac{14}{2}\right)^{2} = (-7)^{2} = 49
\]
2. Add and subtract \(49\) inside the equation:
\[
x^{2} - 14x + 49
\]
### Completing the square for \(y\):
1. Take the coefficient of \(y\) (which is \(2\)), divide it by 2, and square it:
\[
\left(\frac{2}{2}\right)^{2} = 1^{2} = 1
\]
2. Add and subtract \(1\) inside the equation:
\[
y^{2} + 2y + 1
\]
Now, we can rewrite the equation:
\[
(x^{2} - 14x + 49) + (y^{2} + 2y + 1) = -34 + 49 + 1
\]
This simplifies to:
\[
(x - 7)^{2} + (y + 1)^{2} = 16
\]
### Final Standard Form
Thus, the equation of the circle in standard form is:
\[
(x - 7)^{2} + (y + 1)^{2} = 16
\]
This represents a circle with center at \((7, -1)\) and a radius of \(4\) (since \( \sqrt{16} = 4\)).
Quick Answer
The standard form of the circle's equation is \((x - 7)^{2} + (y + 1)^{2} = 16\).
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