Sullivan Bartlett
07/25/2024 · Middle School
The equation of a circle is given in general form. Write the equation of the circle in standard form. \[ x^{2}+y^{2}+6 x-8 y+9=0 \] The standard form of the equation is (Simplify your answer.)
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Step-by-step Solution
Rewrite the equation \( x^{2}+y^{2}+6x-8y+9=0 \).
Identify the conic by following steps:
- step0: Find the standard equation of the circle:
\(x^{2}+y^{2}+6x-8y+9=0\)
- step1: Move the constant to the right side:
\(x^{2}+y^{2}+6x-8y=0-9\)
- step2: Add or subtract the terms:
\(x^{2}+y^{2}+6x-8y=-9\)
- step3: Rearrange the terms:
\(x^{2}+6x+y^{2}-8y=-9\)
- step4: Add the same value to both sides:
\(x^{2}+6x+9+y^{2}-8y=-9+9\)
- step5: Factor the expression:
\(\left(x+3\right)^{2}+y^{2}-8y=-9+9\)
- step6: Add the numbers:
\(\left(x+3\right)^{2}+y^{2}-8y=0\)
- step7: Add the same value to both sides:
\(\left(x+3\right)^{2}+y^{2}-8y+16=16\)
- step8: Rewrite the expression:
\(\left(x+3\right)^{2}+\left(y-4\right)^{2}=16\)
The equation of the circle in standard form is \((x+3)^{2}+(y-4)^{2}=16\).
Quick Answer
\((x+3)^{2}+(y-4)^{2}=16\)
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