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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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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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