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m f = 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 = - 1 m f

Now, we know that the tangent passes through ( x 1 , y 1 ) so the equation is given by:

y - y 1 = m ( x - x 1 ) y - y 1 = - 1 m f ( x - x 1 ) y - y 1 = - 1 y 1 - y 0 x 1 - x 0 ( x - x 1 ) y - y 1 = - x 1 - x 0 y 1 - y 0 ( x - x 1 )

For example, find the equation of the tangent to the circle at point ( 1 , 1 ) . The centre of the circle is at ( 0 , 0 ) . The equation of the circle is x 2 + y 2 = 2 .

Use

y - y 1 = - x 1 - x 0 y 1 - y 0 ( x - x 1 )

with ( x 0 , y 0 ) = ( 0 , 0 ) and ( x 1 , y 1 ) = ( 1 , 1 ) .

y - y 1 = - x 1 - x 0 y 1 - y 0 ( x - x 1 ) y - 1 = - 1 - 0 1 - 0 ( x - 1 ) y - 1 = - 1 1 ( x - 1 ) y = - ( x - 1 ) + 1 y = - x + 1 + 1 y = - x + 2

Co-ordinate geometry

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

Transformations

Rotation of a point about an angle θ

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

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

Polar co-ordinates

Notice that : sin α = y r y = r sin α
and cos α = x r x = r cos α
so P can be expressed in two ways:
  1. P ( x ; y ) rectangular co-ordinates
  2. P ( r cos α ; r sin α ) polar co-ordinates.

Compound angles

(See derivation of formulae in Ch. 12)

sin ( α + β ) = sin α cos β + sin β cos α cos ( α + β ) = cos α cos β - sin α sin β

Now consider P ' After a rotation of θ

P ( x ; y ) = P ( r cos α ; r sin α ) P ' ( r cos ( α + θ ) ; r sin ( α + θ ) )
Expand the co-ordinates of P '
x - co-ordinate of P ' = r cos ( α + θ ) = r cos α cos θ - sin α sin θ = r cos α cos θ - r sin α sin θ = x cos θ - y sin θ
y - co-ordinate of P' = r sin ( α + θ ) = r sin α cos θ + sin θ cos α = r sin α cos θ + r cos α sin θ = y cos θ + x sin θ

which gives the formula P ' = ( x cos θ - y sin θ ; y cos θ + x sin θ ) .

So to find the co-ordinates of P ( 1 ; 3 ) after a rotation of 45 , we arrive at:

P ' = ( x cos θ - y sin θ ) ; ( y cos θ + x sin θ ) = ( 1 cos 45 - 3 sin 45 ) ; ( 3 cos 45 + 1 sin 45 ) = 1 2 - 3 2 ; 3 2 + 1 2 = 1 - 3 2 ; 3 + 1 2

Rotations

Any line O P is drawn (not necessarily in the first quadrant), making an angle of θ degrees with the x -axis. Using the co-ordinates of P and the angle α , calculate the co-ordinates of P ' , if the line O P is rotated about the origin through α degrees.

P α
1. (2, 6) 60
2. (4, 2) 30
3. (5, -1) 45
4. (-3, 2) 120
5. (-4, -1) 225
6. (2, 5) -150

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 A B C with A ( 2 ; 6 ) , B ( 5 ; 6 ) and C ( 5 ; 1 ) . Fill in the lines y = x and y = - x . Complete the table below , by drawing the images of A B C under the given transformations. The first one has been done for you.

Description
Transformation (translation, reflection, Co-ordinates Lengths Angles
rotation, enlargement)
A ' ( 2 ; - 6 ) A ' B ' = 3 B ^ ' = 90
( x ; y ) ( x ; - y ) reflection about the x -axis B ' ( 5 ; - 6 ) B ' C ' = 4 tan A ^ = 4 / 3
C ' ( 5 ; - 2 ) A ' C ' = 5 A ^ = 53 , C ^ = 37
( x ; y ) ( x + 1 ; y - 2 )
( x ; y ) ( - x ; y )
( x ; y ) ( - y ; x )
( x ; y ) ( - x ; - y )
( x ; y ) ( 2 x ; 2 y )
( x ; y ) ( y ; x )
( x ; y ) ( y ; x + 1 )

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

Exercises

  1. Δ A B C undergoes several transformations forming Δ A ' B ' C ' . Describe the relationship between the angles and sides of Δ A B C and Δ A ' B ' C ' (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
    Translate 4 units right
    Enlarge by a scale factor of 2
  2. Δ D E F has E ^ = 30 , D E = 4 cm , E F = 5 cm . Δ D E F is enlarged by a scale factor of 6 to form Δ D ' E ' F ' .
    1. Solve Δ D E F
    2. Hence, solve Δ D ' E ' F '
  3. Δ X Y Z has an area of 6 cm 2 . Find the area of Δ X ' Y ' Z ' if the points have been transformed as follows:
    1. ( x , y ) ( x + 2 ; y + 3 )
    2. ( x , y ) ( y ; x )
    3. ( x , y ) ( 4 x ; y )
    4. ( x , y ) ( 3 x ; y + 2 )
    5. ( x , y ) ( - x ; - y )
    6. ( x , y ) ( x ; - y + 3 )
    7. ( x , y ) ( 4 x ; 4 y )
    8. ( x , y ) ( - 3 x ; 4 y )

Questions & Answers

what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
Smarajit Reply
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Source:  OpenStax, Siyavula textbooks: grade 12 maths. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11242/1.2
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