The tangent (line
) is perpendicular to this line. Therefore,
So,
Now, we know that the tangent passes through
so the equation is given by:
For example, find the equation of the tangent to the circle at point
. The centre of the circle is at
. The equation of the circle is
.
Use
with
and
.
Co-ordinate geometry
Find the equation of the cicle:
with centre
and radius 5
with centre
and radius 4
with centre
and radius 18
with centre
and radius 6
with centre
and radius
Find the equation of the circle with centre
which passes through
.
Where does it cut the line
?
Draw a sketch to illustrate your answers.
Find the equation of the circle with center
which passes through
.
Find the equation of the circle with center
which passes through
.
Find the point where these two circles cut each other.
Find the center and radius of the following circles:
Find the
and
intercepts of the following graphs and draw a sketch to illustrate your answer:
Find the center and radius of the following circles:
Find the equations to the tangent to the circle:
at the point
at the point
at the point
at the point
Transformations
Rotation of a point about an angle
First we will find a formula for the co-ordinates of
after a rotation of
.
We need to know something about polar co-ordinates and compound angles before we start.
Polar co-ordinates
Notice that
:
and
so
can be expressed in two ways:
rectangular co-ordinates
polar co-ordinates.
Compound angles
(See derivation of formulae in Ch. 12)
Now consider
After a rotation of
Expand the co-ordinates of
which gives the formula
.
So to find the co-ordinates of
after a rotation of 45
, we arrive at:
Rotations
Any line
is drawn (not necessarily in the first quadrant), making an angle of
degrees with the
-axis. Using the co-ordinates of
and the angle
, calculate the co-ordinates of
, if the line
is rotated about the origin through
degrees.
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
with
,
and
. Fill in the lines
and
.
Complete the table below , by drawing the images of
under the given transformations. The first one has been done for you.
Description
Transformation
(translation, reflection,
Co-ordinates
Lengths
Angles
rotation, enlargement)
reflection about the
-axis
A transformation that leaves lengths and angles unchanged is called a rigid transformation. Which of the above transformations are rigid?
Exercises
undergoes several transformations forming
. Describe the relationship between the angles and sides of
and
(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
has
,
,
.
is enlarged by a scale factor of 6 to form
.
Solve
Hence, solve
has an area of
. Find the area of
if the points have been transformed as follows:
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