Coordinate Geometry question - help please?

Question

- The line p has gradient 2/3 and passes through the point (6,-12).

- The line q has gradient -1 and passes through the point (5,5).

- The line p meets the y-axis at A

- The line q meets the x-axis at B

Work out the gradient of the line joining points A and B.

Answer 1

Slope intercept form of a line equation y = mx + b

Where m is slope and b is y intercept.

•  The line p has gradient 2/3 and passes through the point (6, -12)

m = 2/3.

Now, the line equation is y = (2/3)x + b.

Find the y - intercept by substituting any point in the line equation say (x, y) = ( 6, -12).

-12 = (2/3)(6) + b

- 12 = 4 + b

b = - 12 - 4

⇒ b = - 16.

The line equation in slope - intercept - form is y = (2/3)x - 16.

• The line q has gradient -1 and passes through the point (5, 5)

m = - 1.

Now, the line equation is y = - x + b.

Find the y - intercept by substituting any point in the line equation say (x, y) = ( 5, 5).

5 = -1(5) + b

5 = - 5 + b

b = 5 + 5

⇒ b = 10.

The line equation in slope - intercept - form is y = - x + 10.

• The line p meets y axis at A

Any line meets y axis at  y intercept.

y = (2/3)x - 16

A = (0, - 16)

• The line q meets x axis at B

Any line meets x axis at x intercept.

y = - x + 10

To find x intercept of above equation substitute y = 0 in y = - x + 10

0 = - x + 10

x = 10

B = (10, 0)

• Let the points are A(x₁, y₁) = (0, - 16) and B(x₂, y₂) = (10, 0).

Gradient of the line joining points A and B (m) = [(y₂ - y₁)/(x₂ -x₁)]

m = [(0 + 16)/(10 - 0)]

m  = 16/10

⇒ m = 8/5.

Gradient of the line joining points A and B = 8/5