Coordinate Geometry - The Circle - created from Mind Map

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Equation of a circle with centre (0,0)

The equation of a circle with centre (0,0) and radius r is x^2+y^2=r^2

To find the equation of a circle, we need to know:

The centre of the circle

The length of the radius.

Finding the radius when given the equation

The circle whose equation is x^2+y^2=r^2 has the centre (0,0) and radius = r.

e.g. the circle x^2+y^2=16, it has

The centre at (0,0)

Radius =(16)^1/2.

Points and Circles

A given point may be inside, on or outside a circle.

A point P is inside a circle if the distance from the centre of the circle to the point P is less than the radius.

A point P is on a circle is the distance from the centre of the circle to the point P is equal to the radius.

A point P is outside a circle if the distance from the centre of the circle to the point P is greater than the radius.

Another method

Another method of determining whether a point is inside, on or outside a circle is to substitute the coordinates of the point into the given equation of the circle.

If x^2+y^2<r^2, the point is inside the circle.

If x^2+y^2=r^2, the point is on the circle.

If x^2+y^2>r^2, the point is outside the circle.

The equation of a circle with centre (h,k) and radius r

The equation of the circle with centre (h,k) and radius r is (x-h)^2+(y-k)^2=r^2.

Finding the centre and radius when given the equation

In the equation (x-h)^2+(y-k)^2=r^2, centre = (h,k) and radius =r.

In the equation (x-2)^2+(y+3)^2=16, we can see that h=2 and k=-3, also r=4.

Intersection of a line and a circle

If a line is a tangent to a circle, it will intersect the circle at one point only.

A circle intersecting the axes

in general, when any line or circle intersects the x-axis, the y-values of the points of intersection will be zero.

Similarly, a line or circle intersects the y-axis at the points where x=0.