12.4 Rotation of Axes

Find a new representation of the given equation after rotating through the given angle.

$3 x^{2} + x y + 3 y^{2} - 5 = 0 , \theta = 45 o$

1. ⓐ $x = - y^{2} - 2 y + 3$
2. ⓑ $9 y^{2} - x^{2} + 18 y - 4 x - 4 = 0$
3. ⓒ $9 x^{2} + 25 y^{2} = 225$
4. ⓓ $x^{2} + y^{2} - 4 x + 10 y - 7 = 0$

Identify the graph of each of the following nondegenerate conic sections.

1. ⓐ $16 y^{2} - x^{2} + x - 4 y - 9 = 0$
2. ⓑ $16 x^{2} + 4 y^{2} + 16 x + 49 y - 81 = 0$

Rewrite the $13 x^{2} - 6 \sqrt{3} x y + 7 y^{2} = 16$ in the $x^{'} y^{'}$ system without the $x^{'} y^{'}$ term.

Identify the conic for each of the following without rotating axes.

1. ⓐ $5 x^{2} + 2 \sqrt{3} x y + 2 y^{2} - 5 = 0$
2. ⓑ $5 x^{2} + 2 \sqrt{3} x y + 12 y^{2} - 5 = 0$

Identify the conic for each of the following without rotating axes.

1. ⓐ $x^{2} - 9 x y + 3 y^{2} - 12 = 0$
2. ⓑ $10 x^{2} - 9 x y + 4 y^{2} - 4 = 0$

What effect does the $x y$ term have on the graph of a conic section?

If the equation of a conic section is written in the form $A x^{2} + B x y + C y^{2} + D x + E y + F = 0 ,$ and $B^{2} - 4 A C > 0 ,$ what can we conclude?

For the equation $A x^{2} + B x y + C y^{2} + D x + E y + F = 0 ,$ the value of $\theta$ that satisfies $\cot \left(2 \theta\right) = \frac{A - C}{B}$ gives us what information?

Determine which conic section is represented based on the given equation.

$x^{2} - 10 x + 4 y - 10 = 0$

Determine which conic section is represented based on the given equation.

$4 x^{2} - y^{2} + 8 x - 1 = 0$

Determine which conic section is represented based on the given equation.

$2 x^{2} + 3 y^{2} - 8 x - 12 y + 2 = 0$

Determine which conic section is represented based on the given equation.

$3 x^{2} + 6 x y + 3 y^{2} - 36 y - 125 = 0$

Determine which conic section is represented based on the given equation.

$2 x^{2} + 4 \sqrt{3} x y + 6 y^{2} - 6 x - 3 = 0$

Determine which conic section is represented based on the given equation.

$8 x^{2} + 4 \sqrt{2} x y + 4 y^{2} - 10 x + 1 = 0$

Find a new representation of the given equation after rotating through the given angle.

$4 x^{2} - x y + 4 y^{2} - 2 = 0 , \theta = 45°$

Find a new representation of the given equation after rotating through the given angle.

$- 2 x^{2} + 8 x y + 1 = 0 , \theta = 45°$

Determine the angle $\theta$ that will eliminate the $x y$ term and write the corresponding equation without the $x y$ term.

$x^{2} + 3 \sqrt{3} x y + 4 y^{2} + y - 2 = 0$

Determine the angle $\theta$ that will eliminate the $x y$ term and write the corresponding equation without the $x y$ term.

$9 x^{2} - 3 \sqrt{3} x y + 6 y^{2} + 4 y - 3 = 0$

Determine the angle $\theta$ that will eliminate the $x y$ term and write the corresponding equation without the $x y$ term.

$16 x^{2} + 24 x y + 9 y^{2} + 6 x - 6 y + 2 = 0$

Determine the angle $\theta$ that will eliminate the $x y$ term and write the corresponding equation without the $x y$ term.

$x^{2} + 4 x y + y^{2} - 2 x + 1 = 0$

Determine the value of $k$ based on the given equation.

Given $2 x^{2} + k x y + 12 y^{2} + 10 x - 16 y + 28 = 0 ,$ find $k$ for the graph to be an ellipse.

Determine the value of $k$ based on the given equation.

Given $k x^{2} + 8 x y + 8 y^{2} - 12 x + 16 y + 18 = 0 ,$ find $k$ for the graph to be a parabola.