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Coordinate Geometry – The Circle
This week the focus is on solving problems which
involve circles, lines meeting circles and lines and
circles intersecting.
Coordinate Geometry – The Circle
CONTENTS:
Tangent to a Circle
Revision Notes
Example 1
Example 2
Example 3
Example 4
Assignment
Coordinate Geometry – The Circle
Tangent to a Circle
The tangent to a circle is a straight line which meets
the circle at one point only.
The angle between the tangent and the radius of the
circle at that point is always 90°, i.e. they are
perpendicular.
Coordinate Geometry – The Circle
Revision Notes
The distance from the centre of a circle to any point
on the circle is the radius.
If we know the coordinates of the centre of a circle
and a point on the circle we can find the radius using
the distance between two points formula:
Distance2 = (x2 – x1)2 + (y2 – y1)2
Coordinate Geometry – The Circle
Revision Notes
The general equation of a circle is given as:
(x – a)2 + (y – b)2 = r2
where (a, b) is the centre of the circle
Coordinate Geometry – The Circle
Revision Notes
If a line and a circle intersect we can find the point(s)
of intersection by solving the equations simultaneously.
With lines and circles we usually use the method of
substitution to solve simultaneously.
That is we substitute the equation of the line into the
equation of the circle.
Coordinate Geometry – The Circle
Example 1:
The line y = 2x + 13 touches the circle x2 + ( y – 3)2 = 20
at (-4, 5). Show that the radius at (-4, 5) is
perpendicular to the line.
Solution:
Click here for solution.
Coordinate Geometry – The Circle
Example 2:
The line y = x – 2 intersects the circle (x + 1)2 + (y + 3)2 =
8 at the points A and B. Find the coordinates of A and B.
Solution:
Click here for solution.
Coordinate Geometry – The Circle
Example 3:
Show that the line 2y + x = 18 is a tangent to the circle
(x – 2)2 + (y – 3)2 = 20 and determine the coordinates of
the point of contact.
Solution:
Click here for solution.
Coordinate Geometry – The Circle
Example 4:
The point P(1, -2) lies on the circle centre (4, 6).
a) Find the equation of the circle
b) Find the equation of the tangent to the circle at P.
Solution:
Click here for solution.
Coordinate Geometry – The Circle
Assignment
This weeks assignment is the discussion forum “Assignment
4: Mixed Problems”.
There are 6 questions from which you choose one. A
maximum of 3 solutions are to be submitted for each
question.
Full instructions are in the forum.
Deadline: 5:00pm on Monday 8 March 2010.