# 8.4 Rotation of axes  (Page 5/8)

 Page 5 / 8

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

1. ${x}^{2}-9xy+3{y}^{2}-12=0$
2. $10{x}^{2}-9xy+4{y}^{2}-4=0$
1. hyperbola
2. ellipse

Access this online resource for additional instruction and practice with conic sections and rotation of axes.

## Key equations

 General Form equation of a conic section $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$ Rotation of a conic section Angle of rotation

## Key concepts

• Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
• A nondegenerate conic section has the general form $\text{\hspace{0.17em}}A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}A,B\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ are not all zero. The values of $\text{\hspace{0.17em}}A,B,$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ determine the type of conic. See [link] .
• Equations of conic sections with an $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term have been rotated about the origin. See [link] .
• The general form can be transformed into an equation in the $\text{\hspace{0.17em}}{x}^{\prime }\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}^{\prime }\text{\hspace{0.17em}}$ coordinate system without the $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ term. See [link] and [link] .
• An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See [link] .

## Verbal

What effect does the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term have on the graph of a conic section?

The $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term causes a rotation of the graph to occur.

If the equation of a conic section is written in the form $\text{\hspace{0.17em}}A{x}^{2}+B{y}^{2}+Cx+Dy+E=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}AB=0,$ what can we conclude?

If the equation of a conic section is written in the form $\text{\hspace{0.17em}}A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,$ and $\text{\hspace{0.17em}}{B}^{2}-4AC>0,$ what can we conclude?

The conic section is a hyperbola.

Given the equation $\text{\hspace{0.17em}}a{x}^{2}+4x+3{y}^{2}-12=0,$ what can we conclude if $\text{\hspace{0.17em}}a>0?$

For the equation $\text{\hspace{0.17em}}A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0,$ the value of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that satisfies $\text{\hspace{0.17em}}\mathrm{cot}\left(2\theta \right)=\frac{A-C}{B}\text{\hspace{0.17em}}$ gives us what information?

It gives the angle of rotation of the axes in order to eliminate the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term.

## Algebraic

For the following exercises, determine which conic section is represented based on the given equation.

$9{x}^{2}+4{y}^{2}+72x+36y-500=0$

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

$AB=0,$ parabola

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

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

$AB=-4<0,$ hyperbola

$4{y}^{2}-5x+9y+1=0$

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

$AB=6>0,$ ellipse

$4{x}^{2}+9xy+4{y}^{2}-36y-125=0$

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

${B}^{2}-4AC=0,$ parabola

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

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

${B}^{2}-4AC=0,$ parabola

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

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

${B}^{2}-4AC=-96<0,$ ellipse

For the following exercises, find a new representation of the given equation after rotating through the given angle.

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

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

$7{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}-4=0$

$2{x}^{2}+8xy-1=0,\theta =30°$

$-2{x}^{2}+8xy+1=0,\theta =45°$

$3{{x}^{\prime }}^{2}+2{x}^{\prime }{y}^{\prime }-5{{y}^{\prime }}^{2}+1=0$

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

For the following exercises, determine the angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that will eliminate the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term and write the corresponding equation without the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term.

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

$\theta ={60}^{\circ },11{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}+\sqrt{3}{x}^{\prime }+{y}^{\prime }-4=0$

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

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

$\theta ={150}^{\circ },21{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}+4{x}^{\prime }-4\sqrt{3}{y}^{\prime }-6=0$

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

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

$\theta \approx {36.9}^{\circ },125{{x}^{\prime }}^{2}+6{x}^{\prime }-42{y}^{\prime }+10=0$

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

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

$\theta ={45}^{\circ },3{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}-\sqrt{2}{x}^{\prime }+\sqrt{2}{y}^{\prime }+1=0$

$4{x}^{2}-2\sqrt{3}xy+6{y}^{2}-1=0$

## Graphical

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.

$y=-{x}^{2},\theta =-{45}^{\circ }$

$\frac{\sqrt{2}}{2}\left({x}^{\prime }+{y}^{\prime }\right)=\frac{1}{2}{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}$

$x={y}^{2},\theta ={45}^{\circ }$

$\frac{{x}^{2}}{4}+\frac{{y}^{2}}{1}=1,\theta ={45}^{\circ }$

$\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{8}+\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}=1$

$\frac{{y}^{2}}{16}+\frac{{x}^{2}}{9}=1,\theta ={45}^{\circ }$

${y}^{2}-{x}^{2}=1,\theta ={45}^{\circ }$

$\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}-\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{2}=1$

$y=\frac{{x}^{2}}{2},\theta ={30}^{\circ }$

$x={\left(y-1\right)}^{2},\theta ={30}^{\circ }$

$\frac{\sqrt{3}}{2}{x}^{\prime }-\frac{1}{2}{y}^{\prime }={\left(\frac{1}{2}{x}^{\prime }+\frac{\sqrt{3}}{2}{y}^{\prime }-1\right)}^{2}$

$\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1,\theta ={30}^{\circ }$

For the following exercises, graph the equation relative to the $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ system in which the equation has no $\text{\hspace{0.17em}}{x}^{\prime }{y}^{\prime }\text{\hspace{0.17em}}$ term.

$xy=9$

${x}^{2}+10xy+{y}^{2}-6=0$

${x}^{2}-10xy+{y}^{2}-24=0$

$4{x}^{2}-3\sqrt{3}xy+{y}^{2}-22=0$

$6{x}^{2}+2\sqrt{3}xy+4{y}^{2}-21=0$

$11{x}^{2}+10\sqrt{3}xy+{y}^{2}-64=0$

$21{x}^{2}+2\sqrt{3}xy+19{y}^{2}-18=0$

$16{x}^{2}+24xy+9{y}^{2}-130x+90y=0$

$16{x}^{2}+24xy+9{y}^{2}-60x+80y=0$

$13{x}^{2}-6\sqrt{3}xy+7{y}^{2}-16=0$

$4{x}^{2}-4xy+{y}^{2}-8\sqrt{5}x-16\sqrt{5}y=0$

For the following exercises, determine the angle of rotation in order to eliminate the $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term. Then graph the new set of axes.

$6{x}^{2}-5\sqrt{3}xy+{y}^{2}+10x-12y=0$

$6{x}^{2}-5xy+6{y}^{2}+20x-y=0$

$\theta ={45}^{\circ }$

$6{x}^{2}-8\sqrt{3}xy+14{y}^{2}+10x-3y=0$

$4{x}^{2}+6\sqrt{3}xy+10{y}^{2}+20x-40y=0$

$\theta ={60}^{\circ }$

$8{x}^{2}+3xy+4{y}^{2}+2x-4=0$

$16{x}^{2}+24xy+9{y}^{2}+20x-44y=0$

$\theta \approx {36.9}^{\circ }$

For the following exercises, determine the value of $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ based on the given equation.

Given $\text{\hspace{0.17em}}4{x}^{2}+kxy+16{y}^{2}+8x+24y-48=0,$ find $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ for the graph to be a parabola.

Given $\text{\hspace{0.17em}}2{x}^{2}+kxy+12{y}^{2}+10x-16y+28=0,$ find $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ for the graph to be an ellipse.

$-4\sqrt{6}

Given $\text{\hspace{0.17em}}3{x}^{2}+kxy+4{y}^{2}-6x+20y+128=0,$ find $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ for the graph to be a hyperbola.

Given $\text{\hspace{0.17em}}k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0,$ find $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ for the graph to be a parabola.

$k=2$

Given $\text{\hspace{0.17em}}6{x}^{2}+12xy+k{y}^{2}+16x+10y+4=0,$ find $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ for the graph to be an ellipse.

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar