Can someone please answer these maths questions with full working (semi easy) Thanx (Analytical Geometry)?

Question

Given the points A( – 1, – 7), B(2, – 1) and C(4, 3):

a. Show that A, B and C are collinear. (Show this algebraically, not by drawing a diagram.)

b. Find the equation of the line joining AC. Answer in the form y = m x + c.

c. State the equation of the line through C that is parallel to the y axis.

d. Calculate the length of AC.

i. State your answer in surdic form, (i.e. using a ).

ii. State your answer as a decimal rounded to 1 decimal place.

Thanx Heaps x*x

Answer 1

a. the vectors AB and AC will have a cross product of 0 if they are collinear.

AB = (2,-1) - (-1,-7) = (3,6)

AC = (4,3) - (-1,-7) = (5,10)

| i   j   k |

| 3  6  0 | = i(6*0 - 10*0) + j(5*0 - 3*0) + k(3*10 - 6*5) = (0,0,0)

| 5 10 0 |

So the cross product is 0 and the points are collinear.

b. slope = [3 - (-7)] / [4- (-1)] = 10/5 = 2 = m

y = (1/2)x + b

Use the point (4,3) to get b. 3 = (2)4 + b and b = 3 - 8 = -5

The line is: y = (2)x - 5

Check with A: -7 = 2(-1) - 5 = -7 and this is good

c. this is just x = 4

d.length AC = SQRT[(4 - (-1))^2 + (3 - (-7))^2] = SQRT(125)

length AC = 5SQRT(5) = 11.2