The full set of rules, formulas and methods — stripped of worked examples — so you can sit and learn the material.
The exact mid–point of the circle.
The centre and radius are the two main parts of a circle.
The distance from the centre to any point on the circle.
A line through the centre from one side of the circle to the other.
A line that hits the circle at two points but does not necessarily go through the centre.
The diameter is a special chord.
A line that touches the outside of the circle at one point only.
x2 + y2 = r2 In Tables
(x − h)2 + (y − k)2 = r2 In Tables
The equation is of the form
x2 + y2 + 2gx + 2fy + c = 0 In Tables
Centre is (−g, −f) — that is, half the coefficient of x and change the sign, half the coefficient of y and change the sign.
Radius = √(g2 + f2 − c)
Sub the co–ordinates of the point (x1, y1) into the LHS of the circle equation and compare to r2.
If (x1)2 + (y1)2 < r2 — point is inside the circle.
If (x1)2 + (y1)2 = r2 — point is on the circle.
If (x1)2 + (y1)2 > r2 — point is outside the circle.
This is done through simultaneous equations as done in algebra — sub the line into the circle.
(a) Two real answers — the line cuts in two places, so the line is a chord to the circle.
(b) One real answer — the line is a tangent.
(c) Two complex answers — the line does not cut the circle.
If the question asks to prove it is a tangent but does not ask for the point of contact, a better way is to show that the perpendicular distance from the line to the centre is equal to the radius.
Three things could happen here:
(i) Perpendicular distance = radius — line is a tangent.
(ii) Perpendicular distance < radius — line is a chord.
(iii) Perpendicular distance > radius — line does not touch.
Step 1. Find the centre of the circle.
Step 2. Find the slope between centre and given point of contact.
Step 3. Invert and change the sign of that slope — that gives the slope of the tangent.
Step 4. Use given point and slope of tangent in the equation of a line:
y − y1 = m(x − x1)
The tangent is perpendicular to the radius at the point of contact.
The equation of the tangent to the circle x2 + y2 = r2 at the point (x1, y1) on the circle is given by
x1x + y1y = r2 In Tables
This is a quicker way of doing the above question — centre must be (0, 0). Proof is given in your proof notes.
Remember a line or circle cuts the
x–axis at y = 0
y–axis at x = 0
If the circle touches the x–axis then g2 = c
If the circle touches the y–axis then f2 = c
The radius will stay the same — only have to transform the centre.
Step 1. Find the centre and radius.
Step 2. Use the transformation to get the new centre.
Step 3. Find the new circle using the new centre and the same old radius:
(x − h)2 + (y − k)2 = r2
A radius (or part of a radius) which is perpendicular to a chord of a circle bisects that chord.
Must find (a) centre, (b) radius and then (c) use the general equation formula.
(a) Centre = midpoint between the two points.
(b) Radius = distance from centre to either point.
(c) Use general equation
(x − h)2 + (y − k)2 = r2
If a triangle is right–angled, then the centre of the circle that passes through the 3 points is the midpoint of the hypotenuse.
Given three points on a circle, sub in each point into
x2 + y2 + 2gx + 2fy + c = 0
to be left with three equations in terms of g, f and c. Solve simultaneously.
Circle is of the form
x2 + y2 + 2gx + 2fy + c = 0
Sub in the two given points.
The centre lies on a certain line, so sub the centre (−g, −f) into that line equation.
Now you have three equations — solve for g, f and c.
If the circle touches the x–axis then g2 = c.
If the circle touches the y–axis then f2 = c.
Combine those conditions with the given point subbed into general form to solve for g, f and c.
The centre (−g, −f) is a certain distance from a given line.
Use the perpendicular distance formula from centre to the line.
Step 1. Let the line be y − y1 = m(x − x1).
Step 2. Sub in the given outside point.
Step 3. Perpendicular distance from centre to the line is equal to the radius.
Step 4. Solve for m (usually two values — two tangents).
If
S = x2 + y2 + 2gx + 2fy + c = 0
and
S1 = x2 + y2 + 2g1x + 2f1y + c1 = 0
represent two circles, then the equation
S − S1 = 0
represents a straight line which is the
(a) common chord if S and S1 have two points of intersection.
(b) common tangent at the point of intersection if S and S1 have only one point of intersection — i.e. they are touching.
Let two circles S1 and S2 have radii r1 and r2, and let d be the distance between their centres.
The circles touch externally if d = r1 + r2.
The circles touch internally if d = |r1 − r2|.
Add radii for outside, subtract radii for inside.