Chapter 10: Q 28. (page 677)

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$x^{2} + 4 x y + 4 y^{2} + 5 \sqrt{5} y + 5 = 0$

Short Answer

Expert verified

$\begin{array}{rcl} x & = & - \frac{3 x^{'}}{5} + \frac{4 y^{'}}{5} \\ y & = & - \frac{4 x^{'}}{5} - \frac{3 y^{'}}{5} \end{array}$

Step by step solution

Step 1. Given information

An equation is $x^{2} + 4 x y + 4 y^{2} + 5 \sqrt{5} y + 5 = 0$ .

Step 2. Determine the appropriate rotation formulas.

The given conic is given by the equation given below.

$x^{2} + 4 x y + 4 y^{2} + 5 \sqrt{5} y + 5 = 0$

The rotation formulae for the given conic such that $x y -$ term is absent are given by the formulae given below, where $θ$ can be determined by the relation $\cot 2 θ = \frac{A - C}{B}$ .

$x = x^{'} \cos θ - y^{'} \sin θ y = x^{'} \sin θ + y^{'} \cos θ$

The value of the $θ$ can be determined by substituting the values of $A, B, C$ in the expression that evaluates the value of angle $θ$ .

width="157">cot2θ=1-44=-342θ=cot-1-34=12cot-1-34

Therefore the rotation formulae for the given equation of conic are given by the equations that follow.

$\begin{array}{rcl} x & = & x^{'} \cos (\cot^{- 1} (\frac{- 3}{4})) - y^{'} \sin (\cot^{- 1} (\frac{- 3}{4})) \\ = & - \frac{3 x^{'}}{5} + \frac{4 y^{'}}{5} \\ y & = & x^{'} \sin (\cot^{- 1} (\frac{- 3}{4})) + y^{'} \cos (\cot^{- 1} (\frac{- 3}{4})) \\ = & - \frac{4 x^{'}}{5} - \frac{3 y^{'}}{5} \end{array}$

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In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$x^{2} + 4 x y + 4 y^{2} + 5 \sqrt{5} y + 5 = 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

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In Problem, determine the appropriate rotation formulas to use so that the new equation contains no xy-term.x2+4xy+4y2+55y+5=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the appropriate rotation formulas.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Discrete Mathematics

Applied Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problem, determine the appropriate rotation formulas to use so that the new equation contains no $x y -$ term.
$x^{2} + 4 x y + 4 y^{2} + 5 \sqrt{5} y + 5 = 0$