Circle Flashcards
the equation of a circle with centre (0,0) and radius r, is:
x squared + y squared = r squared
how to work out radius if given a centre and a point the circle passes through
distance formula
the equation of a circle with centre (a,b) and radius r is:
(x-a) squared + (y-b) squared = r squared
how to work out circle equation if given the points of the diameter AB
find centre of circle (midpoint of AB)
use distance formula to work out radius using centre and either A or B
sub centre and radius into (x-a) squared + (y-b) squared = r squared
given a circle with centre (a,b) and radius r units, how do you determine whether the point (p,q) lies within, outwith or on the circumference?
use (p-a) squared + (q-b) squared = r squared
r squared then outwith circle
what is the general equation of a circle?
x squared + y squared + 2gx + 2fy + c = 0
centre of circle from general equation
(-g,-f)
radius of circle from general equation
square root of g squared + f squared - c
what if g squared + f squared - c < 0
we cannot obtain a real value for the radius, since we would square a negative
not a circle
what if g squared + f squared - c = 0
the radius is zero so the equation represents a point rather than a circle
if a line and a circle only touch at one point, then the line is a…
tangent to the circle at that point
to find out how many times a line and circle meet…
we can use substitution (sub line equation into circle equation)
then find y-coords by subbing x coords into easiest equation (normally line)
to show that a line is a tangent to a circle…
the equation of the line can be substituted into the equation of the circle and solved - there should only be one solution
2 methods to prove that a line is a tangent to a circle
- factorise - (since solutions are equal)
* use method 1 if asks for point of contact/intersection* - discriminant - b squared- 4ac=0
if a line is a tangent to a circle, then the radius…
will meet the tangent at right angles
