P is the point (16,9) and Q is the point (22. 24). (a) Find the equation of the line perpendicular to PQ that passes through the point (5, 1).
Give your answer in the form y=mx+c.
(b) N is the point on PQ such that PN=2NQ. Find the co-ordinates of N.

see explanation

Step-by-step explanation:

(a)

The equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

Calculate m using the slope formula

m =

with (x₁, y₁ ) = (P(16, 9) and (x₂, y₂ ) = Q(22, 24)

= = =

Given a line with slope m then the slope of a line perpendicular to it is

= - = - = - , then

y = - x + c ← is the partial equation

To find c substitute (5, 1) into the partial equation

1 = - 2 + c ⇒ c = 1 + 2 = 3

y = - x + 3 ← equation of perpendicular line

(b)

Given PN = 2NQ , then N sections PQ in the ratio 2 : 1

Using the Section formula to find coordinates of N

N = ( , )

   = ( , )

    = ( , )

    = (20, 19 )

the answer is 0052

0 0 5 2

5 0 2 6 3 5

− 0

2 6

− 0

2 6 3

− 2 5 0

1 3 5

− 1 0 0

3 5