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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 )

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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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