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Chapter 9: Problem 54

Identify each equation without completing the square. $$9 x^{2}+4 y^{2}-36 x+8 y+31=0$$

Short Answer

Expert verified
The given equation represents an ellipse. In standard form, it is \((x-2)^{2}/(4/9) + (y+1)^{2} = 1\).

Step by step solution

01

Rearrange and Group

Rearrange our equation and group the x's and y's together: \(9 (x^{2}-4x) + 4 (y^{2}+2y) = -31\).
02

Analyze

To analyze further we can partially complete the square for each of the variables, this will allow us to identify the type of the conic section: \(9 [(x-2)^{2}-4] + 4 [(y+1)^{2}-1] = -31\) This simplifies to: \(9 (x-2)^{2} + 4 (y+1)^{2} = 4\) The signs of both variable terms are positive, but the coefficients of \(x^{2}\) and \(y^{2}\) are not equal, which indicates that this equation represents an ellipse.
03

Convert to Standard Form

Divide all terms by 4 to convert equation to standard form for ellipse, i.e. of the form \((x-h)^{2}/a^{2} + (y-k)^{2}/b^{2} = 1\): \((x-2)^{2}/(4/9) + (y+1)^{2}/1 = 1\) After simplifying we get: \((x-2)^{2}/(4/9) + (y+1)^{2} = 1\) This equation represents our ellipse in standard form.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to graph an ellipse whose equation contained an \(x y\) -term, I used a rotated coordinate system that placed the ellipse's center at the origin.

In Exercises \(21-40\), eliminate the parameter \(t\). Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of \(t .\) (If an interval for \(t\) is not specified, assume that \(-\infty < t < \infty .)\) $$x=2 \sin t, y=2 \cos t ; 0 \leq t < 2 \pi$$

In Exercises \(53-56,\) find two different sets of parametric equations for each rectangular equation. $$y=4 x-3$$

The following are parametric equations of the line through \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) $$x=x_{1}+t\left(x_{2}-x_{1}\right) \quad \text { and } \quad y=y_{1}+t\left(y_{2}-y_{1}\right)$$ Eliminate the parameter and write the resulting equation in point-slope form.

What does it mean to eliminate the parameter? What useful information can be obtained by doing this?
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