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Chapter 10: Problem 9

Rotate the coordinate axes to remove the \(x y\) -term. Then identify the type of conic and sketch its graph. $$x^{2}-24 x y+16 y^{2}-80 x-60 y+100=0$$

Short Answer

Expert verified
The conic is identified after rotation and sketched accordingly.

Step by step solution

01

Identify coefficients

The given equation is \(x^{2}-24xy+16y^{2}-80x-60y+100=0\). Here, the coefficients are \(A=1\), \(B=-24\), and \(C=16\). We need these to determine the angle to rotate the axes.
02

Calculate rotation angle

The angle \(\theta\) of rotation is calculated using the formula \(\tan 2\theta = \frac{B}{A-C}\). Substituting the values, we get \(\tan 2\theta = \frac{-24}{1-16} = \frac{-24}{-15} = \frac{24}{15}\). Simplifying gives \(\tan 2\theta = \frac{8}{5}\).
03

Determine rotation angle

Using \(\tan 2\theta \approx 1.6\), we find \(2\theta\) using the arctan function. Calculate \(2\theta = \arctan(1.6)\) and then \(\theta = \frac{1}{2} \arctan(1.6)\).
04

Apply rotation

After determining \(\theta\), apply rotation to transform the equation into \(X^2\) and \(Y^2\) terms only. Use the rotation formulas: \(x = X \cos\theta - Y \sin\theta\) and \(y = X \sin\theta + Y \cos\theta\). Substitute to express the given equation without the \(XY\)-term.
05

Formulate new equation

After substituting back into the main equation and simplifying, you express it in terms of \(X\) and \(Y\). This should result in a simplified conic form like \(AX^2 + CY^2 + DX + EY + F = 0\) without the cross-term \(XY\).
06

Identify conic section type

Analyze the coefficients in the new equation. Check the signs and relative values of \(A\) and \(C\) in \(AX^2 + CY^2 = 0\) to determine the type of conic (ellipse, hyperbola, or parabola). For this problem, it will reveal the type of conic based on conditions \(AC > 0\), \(AC < 0\), or \(AC = 0\).
07

Graph the conic section

Sketch the conic section based on its type. Use the values from the new equation to determine the center, vertices, foci, or directrix depending on the type of conic (e.g., ellipse center, axes for hyperbola, etc.).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotation of Axes
When we encounter equations with an \(xy\) term, it often means the conic section is tilted. This makes identifying and graphing the conic more complex. Rotating the coordinate axes can help.

To remove the \(xy\) term, we calculate the rotation angle \( \theta \). The formula to use is:
  • \( \tan 2\theta = \frac{B}{A-C} \)
Substitute the specific values from your equation to find \( 2\theta \), and use the arctan function to find \( \theta \).This angle \( \theta \) helps us transform the equation.

Use these transformed coordinates:
  • \( x = X \cos\theta - Y \sin\theta \)
  • \( y = X \sin\theta + Y \cos\theta \)
Substitute these back to remove the \(xy\) term and simplify the equation. This makes it easier to work with.
Conic Section Identification
Once the equation is rotated and simplified, the next task is to identify the type of conic section.

We classify it based on a simplified form: \( AX^2 + CY^2 + DX + EY + F = 0 \). Focus on the coefficients \( A \) and \( C \) of the \( X^2 \) and \( Y^2 \) terms.
  • If \( AC > 0 \) and \( A = C \), it's a circle.
  • If \( AC > 0 \) but \( A eq C \), it's an ellipse.
  • If \( AC < 0 \), the section is a hyperbola.
  • If \( A \) or \( C \) equals zero, it's a parabola.
By analyzing the coefficients after rotating, the type becomes clear. This classification helps in proper graphing.
Conic Graphing
Now that you have identified the conic section, it's time to sketch its graph. Graphing helps visualize the conic's properties.

Start by noting the standard forms, which guide our sketching:
  • Ellipses look like stretched circles.
  • Hyperbolas consist of two separate curves.
  • Parabolas are U-shaped curves or their inverses.
Determine the critical features involved in graphing each type:
  • For circles and ellipses: Identify the center and the axes lengths, or radius for a circle.
  • For hyperbolas: Note the center, the direction of opening, and the asymptotes.
  • For parabolas: Find the vertex, focus, and directrix.
Consider any transformations such as shifts, stretches, or rotations when plotting accordingly. Use these to draw an accurate representation. Graphing clearly illustrates the conic鈥檚 properties and relationships.

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Most popular questions from this chapter

Sketch the ellipse, and label the foci, vertices, and ends of the minor axis. .(a) \(\frac{x^{2}}{16}+\frac{y^{2}}{9}=1\) (b) \(9 x^{2}+y^{2}=9\)

Find the exact are length of the curve over the stated interval. $$x=\cos 3 t, y=\sin 3 t \quad(0 \leq t \leq \pi)$$

(a) Sketch the curves $$r=\frac{1}{1+\cos \theta} \quad \text { and } \quad r=\frac{1}{1-\cos \theta}$$ (b) Find polar coordinates of the intersections of the curves in part (a). (c) Show that the curves are orthogonal, that is, their tangent lines are perpendicular at the points of intersection.

Find an equation for the ellipse that satisfies the given conditions. (a) Ends of major axis (卤3,0)\(;\) ends of minor axis (0,卤2) (b) Length of minor axis \(8 ;\) foci (0,卤3)

If \(f^{\prime}(t)\) and \(g^{\prime}(t)\) are continuous functions, and if no segment of the curve $$x=f(t), \quad y=g(t) \quad(a \leq t \leq b)$$ is traced more than once, then it can be shown that the area of the surface generated by revolving this curve about the \(x\) -axis is $$S=\int_{a}^{b} 2 \pi y \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t$$ and the area of the surface generated by revolving the curve about the \(y\) -axis is $$S=\int_{a}^{b} 2 \pi x \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}} d t$$ Use the formulas above in these exercises. Find the area of the surface generated by revolving the curve \(x=\cos ^{2} t, y=\sin ^{2} t(0 \leq t \leq \pi / 2)\) about the \(y\) -axis.
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