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Chapter 7: Problem 49

Use the Principal Axes Theorem to perform a rotation of axes to eliminate the \(x y\) -term in the quadratic equation. Identify the resulting rotated conic and give its equation in the new coordinate system. $$x y+x-2 y+3=0$$

Short Answer

Expert verified
The short answer will depend on the specific form of the new equation and its coefficients. The task will require follow-through of the provided method to obtain the results/outputs.

Step by step solution

01

Isolation of the xy term

To start, isolate the \(xy\) term in the equation as follows: \(xy + x - 2y + 3 = 0\) rearranges to \(xy = 2y - x - 3\). Since \(2tan(\theta) = \frac{2y - x}{1}, \theta = tan^{-1}(2y - x)\) as given by Principal Axes Theorem.
02

Performing the rotation

The new coordinates (u, v) are defined as: \[u = x cos(\theta) + y sin(\theta)\] \[v = -x sin(\theta) + y cos(\theta)\] Substitute the values of u and v into the equation and replace \( \theta \) as mentioned in Step 1.
03

Simplifying the new equation

After performing the substitution, the goal is to get rid of the cross-product term. Carefully expand and combine like terms, in order to obtain a simplified equation. The resulting equation represents the conic in its standard form in the rotated coordinate system.
04

Identifying the resulting conic

The resulting conic can be identified by comparing with the standard equations of conics (circle, ellipse, hyperbola, parabola). Notice the presence of both \(u^2\) and \(v^2\) indicates it could be either a circle, ellipse or hyperbola. The identification of the conic type is based on the coefficients of \(u^2\) and \(v^2\) and other contextual clues.

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Most popular questions from this chapter

Find the matrix of the quadratic form associated with the equation. $$16 x^{2}-4 x y+20 y^{2}-72=0$$

For each matrix \(A\), find (if possible) a nonsingular matrix \(P\) such that \(P^{-1} A P\) is diagonal. Verify that \(P^{-1} A P\) is a diagonal matrix with the eigenvalues on the diagonal. $$A=\left[\begin{array}{lll} 4 & 0 & 0 \\ 2 & 2 & 0 \\ 0 & 2 & 2 \end{array}\right]$$

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Determine whether the matrix is orthogonal. $$\left[\begin{array}{rrr} -\frac{4}{5} & 0 & \frac{3}{5} \\ 0 & 1 & 0 \\ \frac{3}{5} & 0 & \frac{4}{5} \end{array}\right]$$
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