Equation problem?

Question

a. Find the equation of the line l which is perpendicular to the line 2x+y=5 and which passes through the point (1,k) where k is a constant

I got the answer for this to be x-1=2y-2k
but i don't know what the second part is asking & how to do it:

b. Hence find the value of k for which the line l passes through the origin.

would appreciate any help...

Answer 1

If the equation is 2x + y = 5, then the slope of the line is -2. To find the slope of a perpendicular line.. take the negative reciprocal of -2 .. which is 1/2.

The new line contains (1, k), so using (x, y) for another point on this line the new equation is...

(y - k) / (x - 1) = 1/2

Which is equivalent to the equation you've given.

Now, if this line intersects the origin, then it passes through the point (0, 0)..

Substitute to get

0 - 1 = 2(0) - 2k

Solve

-1 = -2 k
1/2 = k

So k is 1/2

(which make sense because the slope is 1/2.. if you move right 1, you would have to move up 1/2)

Answer 2

You did (a) right so it makes if fairly simple.

b. Note that dividing both sides by 2 gets you y-k=(1/2)x-(1/2). Substitute x=0 and y= 0 (for the origin is at (0,0)), and you get 0-k=0-(1/2). Divide both sides by negative and you get k=(1/2).

Answer 3

Question a
y = - 2x + 5
perpendicular line has m = 1/2
y - k = (1/2).(x - 1)
2y - 2k = x - 1
- 2k = - 1 (if line passes thro` origin)
k = 1 / 2
y - 1 / 2 = (1/2).(x - 1)
2y - 1 = x - 1
2y - x = 0
y = x / 2

Answer 4

x is 4050
y is 4051