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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

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FARWA Reply
crow and angle have feathers and can fly " it is a connective concept
FARWA
my mother always take decision based on reasoning ; she is practicng realistic thinking .
FARWA
all human are mortal is a formal concept
FARWA
sparrow is a best prototype of reptile
FARWA
echoic memory reflect humans visual system
FARWA Reply
getting the information back from stored area of brain is called retrieval
FARWA
no , echoic memory reflect upon things you hear , that's Iconic memory which deals with visual information
Nancy
I'll be studying psychology soon. Where do I start?
Jane Reply
I would start by learning basic terminology and definitions cause there's a lot of it and learning the different concepts behind psychology and there meanings and there use then go onto introduction into psychology
why is psychology a science?
Godslove Reply
it's a social science
Alex
what we define as science is a systematic arrangement of knowledge based on experimental and observational data.so psychology is a science. as for what kind of science it is depends on how and what information are you using in your approach towards understanding the said data and to what conclusion.
imirror
depending on your approach and understanding Psychology can classify as medical or social science like anthropology
imirror
your friend is under stress because she has not prepared well for the examination what strategies would you suggest her? express in 100 or 150 words
Masi Reply
hello
Masi
yess
dcs
where on the disorders does paranoid thought come in?
Hot Reply
hot shots will give you paranoid thoughts
Kimberly
Kimberly 😂😆😂 yes hot shots will give you paranoid thoughts
Brandon
what compound motivation is attention extension
Gauri Reply
what is psychology
Drishti Reply
The scientific study of the mind
Gina
scientific study of mind and behaviour
Siva
...
Saeed
Scientific study of mind soul behaviour and experiences
Ayush
scientific srtudy of behaviour and cognitive processes
Reshmi
Apply social psychology on real life in Nigerian universities campus
tyosaa Reply
what is the meaning of an idiosyncratic pattern
Krystle Reply
I am here for the first time just here to learn...
michael Reply
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Lisa
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GOPAL
hi,am new here
jennifer
what is this group all about
jennifer
Suppose an individual with OCD experiences obsessive thoughts about germs, contamination, and disease whenever she encounters a doorknob. What might have constituted a viable unconditioned stimulus? 
la Reply
What are factors that influence learning?
Enos Reply
Environment Heredity(I am not sure about heredity)
Tusita
Helpful. .. thanks
Enos
Sure, anytime
Tusita
I have other questions also
Enos
Based on the factors affecting learning, how do we improve learning
Enos
I think that addressing that everyone learns in their own time
James
Peer group influence can also be another reason
Sorie
Also knowing what's going on at home. what pressure are the parents putting on them.
James
am I close or no
James
also is there a lack of care? going one more step. with peer groups do an activity that shows how much they have in common
James
I think one's society has a big role to play in determining what/when & how he/she learns.
BECHEM
multiple reasons at different stages. at the earliest stages of development in a child is behaviour of people around. A baby starts learning basic social expressions and actions by mimicking the people around it. further social and cultural practices enforces learning and behaviour
imirror
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imirror
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imirror
under what schedule of reinforcement do animals learn from ?
Phelisa Reply
the role of emotional intelligence in a courtship behavior
Caren Reply
1
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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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