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