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Chapter 6: Problem 47

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 x^{2}-y^{2}-4 x-3=0$$

Short Answer

Expert verified
The graph of the equation is a Hyperbola.

Step by step solution

01

Identify the terms in the equation

Looking at the given equation \(4x^{2} - y^{2} - 4x - 3 = 0\), there are two squared terms \(-4x^{2}\) and \(-y^{2}\), with different coefficients also using both variables x and y.
02

Reorganize the terms

Rearrange the terms so the squares take one side: \(4x^{2} -4x - (y^{2} + 3) = 0\). Now, complete the square for this expression: \(4(x^{2} - x) - (y^{2} + 3) = 0 \rightarrow 4[(x^{2} - x + 1/4) - 1/4] - (y^{2} + 3) = 0\rightarrow 4[(x - 1/2)^{2} - 1/4] -y^{2} - 3 = 0\). Hence, equation becomes: \(4(x - 1/2)^{2} - y^{2} - 4= 0\). After removing the 4 from the left: \((x - 1/2)^{2} - (y^2 / 4) = 1\).
03

Identify the type of conic section

This equation is of the form \(a^{2}x^{2} - b^{2}y^{2}=1\). This is the standard form of a hyperbola centered at the origin, which opens to the right and left. So, the given equation represents a hyperbola.

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