Can you help me with two math problems?

Question

These two problems were giving me trouble:

Write the expression in the form a + bi:

2 + i√5/3- i√5

Find the value of k if the line joining (4,k) and (6,8) and the line joining (-1,4) and (0,8) are a. parallel and b. perpendicular.

Answer 1

Write the expression in the form a + bi:

2 + i√5/3- i√5

(2 + i√5)/(3 - i√5)

Multiply numerator and denominator by the conjugate of the denominator

[(2 + i√5)(3 + i√5)]/[(3 - i√5)(3 + i√5)]

(1 + 5i√5)/(14)

1/14 + (5i√5)/14

1/14 + [(5√5)/14]i

Find the value of k if the line joining (4,k) and (6,8) and the line joining (-1,4) and (0,8) are a. parallel and b. perpendicular.

The slope of the second line is (8 - 4)/(0 + 1) = 4

A line parallel has the same slope

(8 - k)/(6 - 4) = 4

(8 - k)/2 = 4

8 - k = 8

- k = 0

k = 0

The perpendicular line will have the negative reciprocal of the second line's slope

(8 - k)/(6 - 4) = -1/4

8 - k = - 1/2

- k = - 8 1/2

k = 17/2  

Answer 2

1. Write the expression in the form a + bi:   2 + i√5/3- i√5  :

Factor out i√5 so you get 2 + i√5 ( 1/3 - 1)

2 + i√5 ( -2/3)

=  2 - 2 i√5 / 3

2. If the lines are parallel, then they have the same slope.

(8-4) / (0 -1) = 4 so using the same formula and setting it equal to 4 :

4 = (8-k)/(6-4)

4 = (8-k)/2

cross multiply

8 - k = 8

k = 0

Perpendicular lines have negative reciprocal slopes so the slope would be  - 1/4

-1/4 = (8-k)/2

I'll let you finish that one.

Good luck to you !