# 12.4 Rotation of axes  (Page 2/8)

 Page 2 / 8

## Identifying a conic from its general form

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

1. $4{x}^{2}-9{y}^{2}+36x+36y-125=0$
2. $9{y}^{2}+16x+36y-10=0$
3. $3{x}^{2}+3{y}^{2}-2x-6y-4=0$
4. $-25{x}^{2}-4{y}^{2}+100x+16y+20=0$
1. Rewriting the general form, we have

$A=4\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=-9,$ so we observe that $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ have opposite signs. The graph of this equation is a hyperbola.

2. Rewriting the general form, we have

$A=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=9.\text{\hspace{0.17em}}$ We can determine that the equation is a parabola, since $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is zero.

3. Rewriting the general form, we have

$A=3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=3.\text{\hspace{0.17em}}$ Because $\text{\hspace{0.17em}}A=C,$ the graph of this equation is a circle.

4. Rewriting the general form, we have

$A=-25\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=-4.\text{\hspace{0.17em}}$ Because $\text{\hspace{0.17em}}AC>0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}A\ne C,$ the graph of this equation is an ellipse.

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

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

## Finding a new representation of the given equation after rotating through a given angle

Until now, we have looked at equations of conic sections without an $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term, which aligns the graphs with the x - and y -axes. When we add an $\text{\hspace{0.17em}}xy\text{\hspace{0.17em}}$ term, we are rotating the conic about the origin. If the x - and y -axes are rotated through an angle, say $\text{\hspace{0.17em}}\theta ,$ then every point on the plane may be thought of as having two representations: $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ on the Cartesian plane with the original x -axis and y -axis, and $\text{\hspace{0.17em}}\left({x}^{\prime },{y}^{\prime }\right)\text{\hspace{0.17em}}$ on the new plane defined by the new, rotated axes, called the x' -axis and y' -axis. See [link] .

We will find the relationships between $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ on the Cartesian plane with $\text{\hspace{0.17em}}{x}^{\prime }\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}^{\prime }\text{\hspace{0.17em}}$ on the new rotated plane. See [link] .

The original coordinate x - and y -axes have unit vectors $\text{\hspace{0.17em}}i\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}j\text{\hspace{0.17em}}.$ The rotated coordinate axes have unit vectors $\text{\hspace{0.17em}}{i}^{\prime }\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{j}^{\prime }.$ The angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is known as the angle of rotation    . See [link] . We may write the new unit vectors in terms of the original ones.

Consider a vector $\text{\hspace{0.17em}}u\text{\hspace{0.17em}}$ in the new coordinate plane. It may be represented in terms of its coordinate axes.

Because $\text{\hspace{0.17em}}u={x}^{\prime }{i}^{\prime }+{y}^{\prime }{j}^{\prime },$ we have representations of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ in terms of the new coordinate system.

## Equations of rotation

If a point $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ on the Cartesian plane is represented on a new coordinate plane where the axes of rotation are formed by rotating an angle $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ from the positive x -axis, then the coordinates of the point with respect to the new axes are $\text{\hspace{0.17em}}\left({x}^{\prime },{y}^{\prime }\right).\text{\hspace{0.17em}}$ We can use the following equations of rotation to define the relationship between $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left({x}^{\prime },{y}^{\prime }\right):$

and

Given the equation of a conic, find a new representation after rotating through an angle.

1. Find $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ where and
2. Substitute the expression for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ into in the given equation, then simplify.
3. Write the equations with $\text{\hspace{0.17em}}{x}^{\prime }\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{y}^{\prime }\text{\hspace{0.17em}}$ in standard form.

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
1+cos²A/cos²A=2cosec²A-1
test for convergence the series 1+x/2+2!/9x3
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
Ajith
exponential series
Naveen
what is subgroup
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
e power cos hyperbolic (x+iy)
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
why {2kπ} union {kπ}={kπ}?
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
what is complex numbers
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Solve 2cos x + 3sin x = 0.5