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Chapter 6: Problem 67

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$100 x^{2}+100 y^{2}-100 x+400 y+409=0$$

Short Answer

Expert verified
The graph of the given equation is a circle.

Step by step solution

01

Group \(x\)'s and \(y\)'s

Firstly, the given equation should be rewritten in a way that groups the \(x\) terms and \(y\) terms together: \(100\)(\(x^{2}\)-\(x))+\(100\)(\(y^{2}\)+\(4y))+\(409\)=0. This sets up the equation for completing the square.
02

Completing the Square

To complete the square, we need to take the coefficients of the linear \(x\) and \(y\) terms (in our case -1 and 4), divide them by 2, then square them. For the \(x\) term, we get (-1/2)虏 = 0.25 and for the \(y\) term, we get (4/2)虏 = 4. We add these on both sides of our equation to maintain balance: \(100\)(\(x^{2}\)-\(x\) + 0.25)+\(100\)(\(y^{2}\) +4\(y\)+4)+\(409\)-25-400=0
03

Simplify and Identify

Simplify the formula to standard form: \(100(x-0.5)^2+100(y+2)^2=16\). Finally, divide through by 16 to get the formula in standard form for a circle, which is:\((x-0.5)^2+(y+2)^2=0.16\). It can now be seen that this equation is indeed of a circle, confirming our early prediction.

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