We
may we asked for example:
A Level Maths Notes: C1 – More on Circles
Suppose we have the equation of a circle,
We
may we asked for example:
Find the centre and radius of the circle.
Find if the point
is
inside or outside the the circle.
Find if the line
touches
or intersects the circle.
Find if and where the circle intersects the axes.
To find the radius and centre, we complete the square twice, once
for
and
once for
:
centre
is at
and
radius is 5.
To find if the pointis
inside or outside the circle we can find the distance from the centre
to this point. If the radius, 5, is bigger than this distance, the
point is inside the circle.
thereforeis
inside the circle.
To find if the line(1)
touches or intersects the circle
(2)
we can solve these simultaneous equations. If there are no solutions,
the line does not touch or cross the circle. If there is one
solution, the line touches – is a tangent to – the circle at some
point. If there are two solutions, the line crosses the circle at two
points. Sub (1) into (2).
Use the quadratic formula with
The roots are real and distinct hence the line crosses the circle in two points.
To find out if and where the circle intersects the axes we
put
and
by
turns into the equation of the circle.
Where the circle intersects theaxis,
Use the quadratic formula with
We get two real values here so the circle crosses theaxis
at
Where the circle intersects theaxis,
Use the quadratic formula with
We get two real values here so the circle crosses theaxis
at