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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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 \((x+3)^{2}+(y-4)^{2}=16\).

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\((x+3)^{2}+(y-4)^{2}=16\)
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