Coordinate Geometry

Content

  • Distance Between Two Points
  • Mid Point
  • Gradient of Straight Line
  • Collinear Points
  • Parallel and Perpendicular Lines
  • Equation of Vertical and horizontal Lines
  • Equation of a straight line
  • What You need To Find Equation of a Straight Line?


Distance Between Two Points

The distance between two points (x1,y1) and (x2,y2) is


Coordinate of Midpoint

Find the coordinate of the midpoint between (2 , 3) and (4 , 7).


Gradient of a Straight Line

Find the gradient of a straight line passing through (3, 4) and (5, 9).


Gradient = (y2 – y1) / (x2 – x1)

= (9 – 4) / (5 – 3)

= 5 / 2

= 2.5

Collinear Points

Three Points A,B,C are Collinear, if they lie on the same straight line.

Gradient AB = Gradient BC.


Find y, if A(2 , 3), B(4 , 5) and C(7 , y) are collinear.

If A, B and C are collinear, Gradient AB = Gradient BC.

(5 – 3) / (4 – 2) = (y – 5) /(7 – 4)

2/2 = (y – 5) /3

1 = (y – 5) /3

1 x 3 = (y – 5)

3 = y – 5

3 + 5 = y

y = 8

Parallel and Perpendicular Lines

If two lines are parallel, then they have equal gradients.

If two lines are perpendicular, then multiplication of their gradients equals – 1.

Equation of Vertical and horizontal Lines

Equation of vertical line is x = a, a is x-intercept.

Equation of horizontal line is y = b, b is y-intercept.

Equation of a straight line

y= mx + c

m is the gradient

c is the y-intercept, where the line cuts the y axis.

What You need To Find Equation of a Straight Line?

  • gradient and y-intercept (or)
  • Gradient and coordinate of one point on the straight line (or)
  • Coordinates of two points on the straight line (or)
  • Equation of a Parallel/perpendicular line and
  • y-intercept (or)
  • Equation of a Parallel/perpendicular line and coordinate of one point on the straight line.

Find Equation of Straight Line with gradient is 2 and passing through (3,2).


y = mx + c

y = 2x + c


Since the line passing through (3,2), substitute x = 3 , y = 2

2 = 2(3) + c

c = – 4


Hence the equation of the line is y = 2x – 4.

Find Equation of Straight Line Passing through (2 , 3) and (4, 6).


Gradient = m = (6-3) / (4-2) = 1.5

y = 1.5x + c


Since the line passing through (2 , 3), substitute x = 2 , y = 3

3 = 1.5(2) + c

c = 0


Hence the equation of the line is y = 1.5x.

Find Equation of Straight Line Parallel to y = 5x + 10 and passing through (2 , 3)


Since the lines are parallel, their gradients are equal.

m = 5

y = 5x + c


Since the line passing through (2 , 3), substitute x = 2, y = 3

3 = 5(2) + c

3 = 10 + c

c = – 7


Hence the equation of the line is y = 5x – 7.


Finding Equation of Straight Line Perpendicular to y = – 3x + 7 and passing through (3 , 2)


Since the lines are perpendicular, m = 1/3

y = (1/3) x+ c

Since the line passing through (3,2), substitute x = 3 , y = 2

2 = (1/3) (3) + c

c = 1


Hence the equation of the line is y = (1/3) x + 1

Multiply by 3, 3y = x + 3.

Question

  1. Find the gradient of the line 8y = 4x + 5
  2. Find the value of m if the point (m, m+2) lies on the line 2y = 3x – 9
  3. A(1,3) and B(10,12). Find (a) Length of AB (b) Mid-point of AB (c) Equation of AB.
  4. Find the equation of the line parallel to x-axis and passes through the point (2 , 4).
  5. Find the equation of the line parallel to y-axis and passes through the point (–6 , –3).
  6. Find the equation of the line that is parallel to y = 2x + 5 and passes through the point (1 , 5).
  7. Find the equation of the line that is parallel to y = –6x + 12 and passes through the point (6 , 2).