Circles

Equation of a circle (centre 0,0)

x²+y² =r²

Equation of a circle(different centre)

(x-a)² +(y-b)²

-centre is (a,b)

How do we find the equations using points

We need centre and radius

• we find centre thought the midpoint of diameter

• We then find the distance (Pythagoras method like in straight lines) between centre and another point for radius

How can we use completing the square

To find radius and centre of circles

• If we complete the square of messy equation( both for x and y) it ends up being in the form of a circle equations

What must we do when completing the square for circle equations

Collect / rearrange to bring x and y terms together

What must be true for r² for circle equation to work

Bigger than 0

How do we deal with intersections of lines and circles

-solve the equations simultaneously

to find point of intersection

-can also use the discriminant to find out how many times it intersect

What can we do to make dealing with a hard quadratic/substitution easier

factor out

-recognise still what a,b and c are (they might be unknowns e.g k)

What are the circle theorems that we use in this chapter

-tangents and radius are perpendicular

-perpendicular bisector of any chord always passes through the centre of circle

-Right angled triangle in a semi circle

-bisectors of inscribed triangles meet at the centre

How can we use the fact that the tangent and radius are always perpendicular

We can find the equation of the tangent(vice versa)

-find gradient of r and use the negative reciprocal for gradient of the tangent( can also use tangent to find r)

-use a point that lies on the line to substitute

How can we find equation of tangent with just its gradient and equation of the circle

-with equation of the circle we have its centre

-find equation of line that passes through centre ( r )

-the gradient of that line(radius) is negative reciprocal of the tangent as they are perpendicular

-find intersections of radius and tangents to get a point the tangent touches

-now can find equation(s) of tangent by substituting this point into the tangent equation.

When we are finding an equation of a line what do we need

a point it passes through and the gradient

When finding equation of circle what do we need

centre and radius