# 7.1 Coordinate geometry, equation of tangent, transformations  (Page 2/2)

 Page 2 / 2
${m}_{f}=\frac{{y}_{1}-{y}_{0}}{{x}_{1}-{x}_{0}}$

The tangent (line $g$ ) is perpendicular to this line. Therefore,

${m}_{f}×{m}_{g}=-1$

So,

${m}_{g}=-\frac{1}{{m}_{f}}$

Now, we know that the tangent passes through $\left({x}_{1},{y}_{1}\right)$ so the equation is given by:

$\begin{array}{ccc}\hfill y-{y}_{1}& =& m\left(x-{x}_{1}\right)\hfill \\ \hfill y-{y}_{1}& =& -\frac{1}{{m}_{f}}\left(x-{x}_{1}\right)\hfill \\ \hfill y-{y}_{1}& =& -\frac{1}{\frac{{y}_{1}-{y}_{0}}{{x}_{1}-{x}_{0}}}\left(x-{x}_{1}\right)\hfill \\ \hfill y-{y}_{1}& =& -\frac{{x}_{1}-{x}_{0}}{{y}_{1}-{y}_{0}}\left(x-{x}_{1}\right)\hfill \end{array}$

For example, find the equation of the tangent to the circle at point $\left(1,1\right)$ . The centre of the circle is at $\left(0,0\right)$ . The equation of the circle is ${x}^{2}+{y}^{2}=2$ .

Use

$y-{y}_{1}=-\frac{{x}_{1}-{x}_{0}}{{y}_{1}-{y}_{0}}\left(x-{x}_{1}\right)$

with $\left({x}_{0},{y}_{0}\right)=\left(0,0\right)$ and $\left({x}_{1},{y}_{1}\right)=\left(1,1\right)$ .

$\begin{array}{ccc}\hfill y-{y}_{1}& =& -\frac{{x}_{1}-{x}_{0}}{{y}_{1}-{y}_{0}}\left(x-{x}_{1}\right)\hfill \\ \hfill y-1& =& -\frac{1-0}{1-0}\left(x-1\right)\hfill \\ \hfill y-1& =& -\frac{1}{1}\left(x-1\right)\hfill \\ \hfill y& =& -\left(x-1\right)+1\hfill \\ \hfill y& =& -x+1+1\hfill \\ \hfill y& =& -x+2\hfill \end{array}$

## Co-ordinate geometry

1. Find the equation of the cicle:
1. with centre $\left(0;5\right)$ and radius 5
2. with centre $\left(2;0\right)$ and radius 4
3. with centre $\left(5;7\right)$ and radius 18
4. with centre $\left(-2;0\right)$ and radius 6
5. with centre $\left(-5;-3\right)$ and radius $\sqrt{3}$
1. Find the equation of the circle with centre $\left(2;1\right)$ which passes through $\left(4;1\right)$ .
2. Where does it cut the line $y=x+1$ ?
1. Find the equation of the circle with center $\left(-3;-2\right)$ which passes through $\left(1;-4\right)$ .
2. Find the equation of the circle with center $\left(3;1\right)$ which passes through $\left(2;5\right)$ .
3. Find the point where these two circles cut each other.
2. Find the center and radius of the following circles:
1. ${\left(x-9\right)}^{2}+{\left(y-6\right)}^{2}=36$
2. ${\left(x-2\right)}^{2}+{\left(y-9\right)}^{2}=1$
3. ${\left(x+5\right)}^{2}+{\left(y+7\right)}^{2}=12$
4. ${\left(x+4\right)}^{2}+{\left(y+4\right)}^{2}=23$
5. $3{\left(x-2\right)}^{2}+3{\left(y+3\right)}^{2}=12$
6. ${x}^{2}-3x+9={y}^{2}+5y+25=17$
3. Find the $x-$ and $y-$ intercepts of the following graphs and draw a sketch to illustrate your answer:
1. ${\left(x+7\right)}^{2}+{\left(y-2\right)}^{2}=8$
2. ${x}^{2}+{\left(y-6\right)}^{2}=100$
3. ${\left(x+4\right)}^{2}+{y}^{2}=16$
4. ${\left(x-5\right)}^{2}+{\left(y+1\right)}^{2}=25$
4. Find the center and radius of the following circles:
1. ${x}^{2}+6x+{y}^{2}-12y=-20$
2. ${x}^{2}+4x+{y}^{2}-8y=0$
3. ${x}^{2}+{y}^{2}+8y=7$
4. ${x}^{2}-6x+{y}^{2}=16$
5. ${x}^{2}-5x+{y}^{2}+3y=-\frac{3}{4}$
6. ${x}^{2}-6nx+{y}^{2}+10ny=9{n}^{2}$
5. Find the equations to the tangent to the circle:
1. ${x}^{2}+{y}^{2}=17$ at the point $\left(1;4\right)$
2. ${x}^{2}+{y}^{2}=25$ at the point $\left(3;4\right)$
3. ${\left(x+1\right)}^{2}+{\left(y-2\right)}^{2}=25$ at the point $\left(3;5\right)$
4. ${\left(x-2\right)}^{2}+{\left(y-1\right)}^{2}=13$ at the point $\left(5;3\right)$

## Rotation of a point about an angle $\theta$

First we will find a formula for the co-ordinates of $P$ after a rotation of $\theta$ .

We need to know something about polar co-ordinates and compound angles before we start.

## Polar co-ordinates

 Notice that : $sin\alpha =\frac{y}{r}$ $\therefore y=rsin\alpha$ and $cos\alpha =\frac{x}{r}$ $\therefore x=rcos\alpha$
so $P$ can be expressed in two ways:
1. $P\left(x;y\right)$ rectangular co-ordinates
2. $P\left(rcos\alpha ;rsin\alpha \right)$ polar co-ordinates.

## Compound angles

(See derivation of formulae in Ch. 12)

$\begin{array}{ccc}\hfill sin\left(\alpha +\beta \right)& =& sin\alpha cos\beta +sin\beta cos\alpha \hfill \\ \hfill cos\left(\alpha +\beta \right)& =& cos\alpha cos\beta -sin\alpha sin\beta \hfill \end{array}$

## Now consider ${P}^{\text{'}}$ After a rotation of $\theta$

$\begin{array}{ccc}& & P\left(x;y\right)=P\left(rcos\alpha ;rsin\alpha \right)\hfill \\ & & {P}^{\text{'}}\left(rcos\left(\alpha +\theta \right);rsin\left(\alpha +\theta \right)\right)\hfill \end{array}$
Expand the co-ordinates of ${P}^{\text{'}}$
$\begin{array}{ccc}\hfill x-\mathrm{co-ordinate of}{P}^{\text{'}}& =& rcos\left(\alpha +\theta \right)\hfill \\ & =& r\left[cos,\alpha ,cos,\theta ,-,sin,\alpha ,sin,\theta \right]\hfill \\ & =& rcos\alpha cos\theta -rsin\alpha sin\theta \hfill \\ & =& xcos\theta -ysin\theta \hfill \end{array}$
$\begin{array}{ccc}\hfill y-\mathrm{co-ordinate of P\text{'}}& =& rsin\left(\alpha +\theta \right)\hfill \\ & =& r\left[sin,\alpha ,cos,\theta ,+,sin,\theta ,cos,\alpha \right]\hfill \\ & =& rsin\alpha cos\theta +rcos\alpha sin\theta \hfill \\ & =& ycos\theta +xsin\theta \hfill \end{array}$

which gives the formula ${P}^{\text{'}}=\left[\left(,x,cos,\theta ,-,y,sin,\theta ,;,y,cos,\theta ,+,x,sin,\theta ,\right)\right]$ .

So to find the co-ordinates of $P\left(1;\sqrt{3}\right)$ after a rotation of 45 ${}^{\circ }$ , we arrive at:

$\begin{array}{ccc}\hfill {P}^{\text{'}}& =& \left[\left(,x,cos,\theta ,-,y,sin,\theta ,\right),;,\left(,y,cos,\theta ,+,x,sin,\theta ,\right)\right]\hfill \\ & =& \left[\left(1cos{45}^{\circ }-\sqrt{3}sin{45}^{\circ }\right),;,\left(\sqrt{3}cos{45}^{\circ }+1sin{45}^{\circ }\right)\right]\hfill \\ & =& \left[\left(\frac{1}{\sqrt{2}},-,\frac{\sqrt{3}}{\sqrt{2}}\right),;,\left(\frac{\sqrt{3}}{\sqrt{2}},+,\frac{1}{\sqrt{2}}\right)\right]\hfill \\ & =& \left(\frac{1-\sqrt{3}}{\sqrt{2}},;,\frac{\sqrt{3}+1}{\sqrt{2}}\right)\hfill \end{array}$

## Rotations

Any line $OP$ is drawn (not necessarily in the first quadrant), making an angle of $\theta$ degrees with the $x$ -axis. Using the co-ordinates of $P$ and the angle $\alpha$ , calculate the co-ordinates of ${P}^{\text{'}}$ , if the line $OP$ is rotated about the origin through $\alpha$ degrees.

 $P$ $\alpha$ 1. (2, 6) 60 ${}^{\circ }$ 2. (4, 2) 30 ${}^{\circ }$ 3. (5, -1) 45 ${}^{\circ }$ 4. (-3, 2) 120 ${}^{\circ }$ 5. (-4, -1) 225 ${}^{\circ }$ 6. (2, 5) -150 ${}^{\circ }$

## Characteristics of transformations

Rigid transformations like translations, reflections, rotations and glide reflections preserve shape and size, and that enlargement preserves shape but not size.

## Geometric transformations:

Draw a large 15 $×$ 15 grid and plot $▵ABC$ with $A\left(2;6\right)$ , $B\left(5;6\right)$ and $C\left(5;1\right)$ . Fill in the lines $y=x$ and $y=-x$ . Complete the table below , by drawing the images of $▵ABC$ under the given transformations. The first one has been done for you.

 Description Transformation (translation, reflection, Co-ordinates Lengths Angles rotation, enlargement) ${A}^{\text{'}}\left(2;-6\right)$ ${A}^{\text{'}}{B}^{\text{'}}=3$ ${\stackrel{^}{B}}^{\text{'}}={90}^{\circ }$ $\left(x;y\right)\to \left(x;-y\right)$ reflection about the $x$ -axis ${B}^{\text{'}}\left(5;-6\right)$ ${B}^{\text{'}}{C}^{\text{'}}=4$ $tan\stackrel{^}{A}=4/3$ ${C}^{\text{'}}\left(5;-2\right)$ ${A}^{\text{'}}{C}^{\text{'}}=5$ $\therefore \stackrel{^}{A}={53}^{\circ },\stackrel{^}{C}={37}^{\circ }$ $\left(x;y\right)\to \left(x+1;y-2\right)$ $\left(x;y\right)\to \left(-x;y\right)$ $\left(x;y\right)\to \left(-y;x\right)$ $\left(x;y\right)\to \left(-x;-y\right)$ $\left(x;y\right)\to \left(2x;2y\right)$ $\left(x;y\right)\to \left(y;x\right)$ $\left(x;y\right)\to \left(y;x+1\right)$

A transformation that leaves lengths and angles unchanged is called a rigid transformation. Which of the above transformations are rigid?

## Exercises

1. $\Delta ABC$ undergoes several transformations forming $\Delta {A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ . Describe the relationship between the angles and sides of $\Delta ABC$ and $\Delta {A}^{\text{'}}{B}^{\text{'}}{C}^{\text{'}}$ (e.g., they are twice as large, the same, etc.)
 Transformation Sides Angles Area Reflect Reduce by a scale factor of 3 Rotate by 90 ${}^{\circ }$ Translate 4 units right Enlarge by a scale factor of 2
2. $\Delta DEF$ has $\stackrel{^}{E}={30}^{\circ }$ , $DE=4\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ , $EF=5\phantom{\rule{0.166667em}{0ex}}\mathrm{cm}$ . $\Delta DEF$ is enlarged by a scale factor of 6 to form $\Delta {D}^{\text{'}}{E}^{\text{'}}{F}^{\text{'}}$ .
1. Solve $\Delta DEF$
2. Hence, solve $\Delta {D}^{\text{'}}{E}^{\text{'}}{F}^{\text{'}}$
3. $\Delta XYZ$ has an area of $6\phantom{\rule{0.166667em}{0ex}}{\mathrm{cm}}^{2}$ . Find the area of $\Delta {X}^{\text{'}}{Y}^{\text{'}}{Z}^{\text{'}}$ if the points have been transformed as follows:
1. $\left(x,y\right)\to \left(x+2;y+3\right)$
2. $\left(x,y\right)\to \left(y;x\right)$
3. $\left(x,y\right)\to \left(4x;y\right)$
4. $\left(x,y\right)\to \left(3x;y+2\right)$
5. $\left(x,y\right)\to \left(-x;-y\right)$
6. $\left(x,y\right)\to \left(x;-y+3\right)$
7. $\left(x,y\right)\to \left(4x;4y\right)$
8. $\left(x,y\right)\to \left(-3x;4y\right)$

what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!