C1 – Coordinate Geometry Summary
 The midpoint M of points A
C1 – Coordinate Geometry Summary
x1 , y1  and B x2 , y 2  is given by
 x1  x 2 y1  y 2 
,
.
2 
 2
M
 The distance between points A
AB =
x 2  x1 2   y 2  y1 2
x1 , y1  and B x2 , y 2  is given by
.
x1 , y1  and B x2 , y 2  is given by
y 2  y1
.
x 2  x1
 x1  x 2 y1  y 2 
,
.
2 
 2
 The distance between points A
AB =
y  y1  mx  x1  where m is the gradient and x1 , y1  is a point
on the line.
3. ax  by  c  0 - a rearrangement of the equations above.
 The equation of a circle with centre
a, b and radius r is given by
x 2  x1 2   y 2  y1 2
x1 , y1  and B x2 , y 2  is given by
.
 The gradient of the line through A
m AB 
 When two straight lines are perpendicular the product of their gradients
is -1.
 The general equation of a straight line is given by
1. y  mx  c where m is the gradient and c is the y-intercept.
2.
x1 , y1  and B x2 , y 2  is given by
M
 The gradient of the line through A
m AB 
 The midpoint M of points A
x1 , y1  and B x2 , y 2  is given by
y 2  y1
.
x 2  x1
 When two straight lines are perpendicular the product of their gradients
is -1.
 The general equation of a straight line is given by
1. y  mx  c where m is the gradient and c is the y-intercept.
2.
y  y1  mx  x1  where m is the gradient and x1 , y1  is a point
on the line.
3. ax  by  c  0 - a rearrangement of the equations above.
 The equation of a circle with centre
a, b and radius r is given by
x  a 2   y  b2  r 2 . This is often expanded and rearranged to
x  a 2   y  b2  r 2 . This is often expanded and rearranged to
equal 0.
 Properties of circles
1. The angle in a semi-circle is 90.
2. The radius perpendicular to a chord bisects the chord.
3. The tangent to a circle at point P touches the circle at P only.
4. The normal to a circle at point P is perpendicular to the tangent at
point P and passes through the centre.
5. Two tangents to a circle from a point outside the circle are equal in
length.
 Intersection of a line and a circle
Solve the equations simultaneously to get a quadratic equation in x.
1. Intersect at two points: quadratic equation has two distinct real roots,
equal 0.
 Properties of circles
1. The angle in a semi-circle is 90.
2. The radius perpendicular to a chord bisects the chord.
3. The tangent to a circle at point P touches the circle at P only.
4. The normal to a circle at point P is perpendicular to the tangent at
point P and passes through the centre.
5. Two tangents to a circle from a point outside the circle are equal in
length.
 Intersection of a line and a circle
Solve the equations simultaneously to get a quadratic equation in x.
1. Intersect at two points: quadratic equation has two distinct real roots,
b 2  4ac  0
b 2  4ac  0
2. Intersect at one point: quadratic equation has one repeated root,
b 2  4ac  0
3. Do not intersect: quadratic equation has no real roots,
2. Intersect at one point: quadratic equation has one repeated root,
b 2  4ac  0
The line is tangent to the circle.
b  4ac  0
2
The line is tangent to the circle.
3. Do not intersect: quadratic equation has no real roots,
b 2  4ac  0