Circles

Question 1

Circle equation

Answer 1

(x-a)^2 +(y-b)^2 = r^2

where r is radius, and a and b are x and y in the coordinates for the centre of the circle

Question 2

Testing points

Answer 2

For circle with (a,b) centre and radius r, point (p,q) will be …
(p-a)^2 + (q-b)^2 < r^2 will lie within the circle
(p-a)^2 + (q-b)^2 = r^2 will lie on the circumference of the circle
(p-a)^2 + (q-b)^2 > r^2 will lie outside the circle

Question 3

General equation of a circle

Answer 3

x^2 +y^2 + 2gx +2fy + c =0

Question 4

Centre and radius equations for general equation of a circle

Answer 4

Centre (-g,-f)

Radius = square root of g^2 + f^2 -c units

Question 5

Intersection of line and circle

Answer 5

Substitute equation of line into equation of circle (sub it in wherever y is)
Now solve this equation to find x values of intersection, now substitue the x value(s) into the straight line equation to find y values of points of intersection

Question 6

How to prove a line is tangent to a circle

Answer 6

Similar process for intersection of lines just show there is only one value of x so only one point of intersection

Question 7

How to find equation for tangent of a circle with point of tangent

Answer 7

Find centre of the circle, now find gradient of the line between that and point of tangent. Now use m*mtan = -1 for m of tangent.
Now use one of these points as a and b and m in the equation y-b = m(x-a)

Question 8

Intersection of circles

Answer 8

Find r1 and r2
Now find d between two centres by using the equation d = square root of (y2-y1)^2 + (x1-x2)^2
If ; d> r1 +r2 circles do not touch
     d= r1 +r2 circles touch externally 
     r1-r2