Chapter 10: Q32. (page 677)

In Problems 31– 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.
$x^{2} - 4 x y + y^{2} - 3 = 0$

Short Answer

Expert verified

$x = \frac{1}{\sqrt{2}} x' - \frac{1}{\sqrt{2}} y' y = \frac{1}{\sqrt{2}} x' + \frac{1}{\sqrt{2}} y'$

Step by step solution

Step 1. Given Information

The given equation is $x^{2} - 4 x y + y^{2} - 3 = 0$ .

We need to determine the appropriate rotation formula so that new equation does not contain $x y$ term.

Step 2. Finding acute angle

Here $A = 1, B = - 4, C = 1$

localid="1646901593711" $c o t 2 θ = \frac{1 - 1}{4} = 0$

Where

$0^{\circ} < 2 θ < 180^{\circ}$

localid="1646901649612" $c o t 2 θ = 0 2 θ = 90 θ = 45^{\circ}$

Step 3. Finding the value of x and y

$x = x' \cos 45^{\circ} - y' \sin 45^{\circ} x = \frac{1}{\sqrt{2}} x' - \frac{1}{\sqrt{2}} y'$

and

$y = x' \sin 45^{\circ} + y' \cos 45^{\circ} y = \frac{1}{\sqrt{2}} x' + \frac{1}{\sqrt{2}} y'$

Step 4. Substituting these values into the original equation

$x^{2} - 4 x y + y^{2} - 3 = 0 \frac{1}{2} {(x' - y')}^{2} - 4 [\frac{1}{\sqrt{2}} (x' - y')] [\frac{1}{\sqrt{2}} (x' + y')] + \frac{1}{2} {(x' + y')}^{2} - 3 = 0 \frac{1}{2} (x'^{2} - 2 x' y' + y'^{2}) - 2 (x'^{2} - y'^{2}) + \frac{1}{2} (x'^{2} + 2 x' y' + y'^{2}) = 3$

Step 5. Simplify above expression

$\frac{1}{2} (x'^{2} - 2 x' y' + y'^{2}) - 2 (x'^{2} - y'^{2}) + \frac{1}{2} (x'^{2} + 2 x' y' + y'^{2}) = 3 \frac{1}{2} x'^{2} - x' y' + \frac{1}{2} y'^{2} - 2 x'^{2} + 2 y'^{2} + \frac{1}{2} x'^{2} + x' y' + \frac{1}{2} y'^{2} = 3 \frac{1}{2} x'^{2} + \frac{1}{2} x'^{2} - x' y' + x' y' + \frac{1}{2} y'^{2} + \frac{1}{2} y'^{2} - 2 x'^{2} + 2 y'^{2} = 3 x'^{2} + y'^{2} - 2 x'^{2} + 2 y'^{2} = 3 - x'^{2} + 3 y'^{2} = 3 \frac{y'^{2}}{1} - \frac{x'^{2}}{3} = 1$

This equation represent hyperbola.

Step 6. Graphing Equation

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In Problems 31– 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.
$x^{2} - 4 x y + y^{2} - 3 = 0$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding the value of x and y

Step 4. Substituting these values into the original equation

Step 5. Simplify above expression

Step 6. Graphing Equation

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In Problems 31– 42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.x2-4xy+y2-3=0

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding acute angle

Step 3. Finding the value of x and y

Step 4. Substituting these values into the original equation

Step 5. Simplify above expression

Step 6. Graphing Equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Statistics

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 31– 42, rotate the axes so that the new equation contains no $x y$ -term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31– 40.
$x^{2} - 4 x y + y^{2} - 3 = 0$