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Chapter 6: Problem 64

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 y^{2}+4 x^{2}-24 x+35=0$$

Short Answer

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

Step by step solution

01

Rewrite The Equation

Rewrite the given equation: \(4 y^{2}+4 x^{2}-24 x+35=0\) in a way that brings out clearly the coefficients of \(y^2\) and \(x^2\) terms. We'll do this by isolating the terms with \(x\) on one side to get: \(4 x^{2} - 24x + 4 y^{2} = -35\).
02

Complete The Square

Complete the square for the terms involving x. Doing this allows you to rewrite the quadratic expression on the left-hand side in square form. Rearrange the equation as \(4(x^{2} - 6x) + 4 y^{2} = -35\). Now, add 9 to both sides to complete the square for the expressions involving x, giving you \(4(x - 3)^{2} + 4 y^{2} = -35 + 36 = 1\).
03

Evaluate The Equation

Divide the whole equation by 4 to normalize the coefficients of the squares: \((x - 3)^{2} + y^{2} = 1/4\). Now, it can be clearly seen that the equation falls into the form of a circle with a radius of 1/2 and whose center is at (3, 0).

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