Coordinate Geometry Worked Examples for IGCSE Maths

Question

Find the equation of the line passing through (1, 3) and (5, 11).

Solution

1. 1

   Find the gradient

   m = (11-3)/(5-1) = 8/4 = 2

   Gradient = change in y / change in x.

2. 2

   Use y = mx + c

   y = 2x + c. Substitute (1, 3): 3 = 2(1) + c, so c = 1.

   Substitute one point to find c.

3. 3

   Write the equation

   y = 2x + 1

   The complete equation of the line.

Question

Find the midpoint and the distance between A(3, -2) and B(9, 6).

Solution

1. 1

   Find the midpoint

   Midpoint = ((3+9)/2, (-2+6)/2) = (6, 2)

   Average the x-coordinates and average the y-coordinates.

2. 2

   Find the distance

   d = sqrt((9-3)^2 + (6-(-2))^2) = sqrt(36 + 64) = sqrt(100) = 10

   Use the distance formula, which is Pythagoras in coordinate form.

Question

Find the equation of the perpendicular bisector of the line segment joining A(2, 1) and B(6, 9).

Solution

1. 1

   Find the midpoint

   Midpoint = ((2+6)/2, (1+9)/2) = (4, 5)

   The perpendicular bisector passes through the midpoint.

2. 2

   Find the gradient of AB

   m(AB) = (9-1)/(6-2) = 8/4 = 2

   Calculate the gradient of the original line segment.

3. 3

   Find the perpendicular gradient

   m(perp) = -1/2

   The negative reciprocal of 2 is -1/2.

4. 4

   Write the equation

   y - 5 = -1/2(x - 4), so y = -1/2 x + 2 + 5 = -1/2 x + 7

   Use point-gradient form with the midpoint and perpendicular gradient.

Question

Find the points of intersection of y = x + 2 and y = x squared - 2x.

Solution

1. 1

   Set the equations equal

   x + 2 = x squared - 2x

   At intersection points, the y-values are equal.

2. 2

   Rearrange to standard form

   0 = x squared - 3x - 2

   Move all terms to one side.

3. 3

   Solve using the quadratic formula

   x = (3 +/- sqrt(9+8))/2 = (3 +/- sqrt(17))/2. x = 3.56 or x = -0.56 (2 d.p.)

   Since it does not factorise neatly, use the formula.

4. 4

   Find y-coordinates

   When x = 3.56: y = 3.56 + 2 = 5.56. When x = -0.56: y = -0.56 + 2 = 1.44.

   Substitute x-values into the simpler equation.