coordinate geometry

[sana.noor]

(77) In an xy-coordinate plane, a line is defined by y = kx + 1. If (4, b), (a, 4), and (a, b +1) are three points on the line, where a and b are unknown, then k = ?

a) 1/2
b) 1
c) 1 and 1/2
d) 2
e) 2 and 1/2

OA is A

[ganeshrkamath]

(77) In an xy-coordinate plane, a line is defined by y = kx + 1. If (4, b), (a, 4), and (a, b +1) are three points on the line, where a and b are unknown, then k = ?

a) 1/2
b) 1
c) 1 and 1/2
d) 2
e) 2 and 1/2

OA is A

y = kx + 1

We know three points on the line : (4,b), (a,4), and (a,b+1)

b = k4 + 1__________________(1)
4 = ak + 1__________________(2)
b+1 = ak + 1______________(3)

(2) - (3) => 4 - (b+1) = 0
3 - b = 0
b = 3

Put this in (1):
3 = 4k + 1
4k = 2
[spoiler]k = 1/2[/spoiler]

Cheers

[Brent@GMATPrepNow]

(77) In an xy-coordinate plane, a line is defined by y = kx + 1. If (4, b), (a, 4), and (a, b +1) are three points on the line, where a and b are unknown, then k = ?

a) 1/2
b) 1
c) 1 and 1/2
d) 2
e) 2 and 1/2

There's a HUGE CLUE in the fact that (a, 4) and (a, b+1) are both on the same line. Notice that the x-coordinates are the same. If the x-coordinates are the same, then there are two possible scenarios:
scenario #1: The points (a, 4) and (a, b+1) are DIFFERENT points, in which case the line is vertical (with undefined slope)
scenario #2: The points (a, 4) and (a, b+1) define the SAME point

IMPORTANT: If a line is defined by y = kx + 1, then k represents the slope. So, the question is really asking us to find the slope of the line.

In scenario #1, the slope would be undefined. Since none of the answer choices are undefined, we can rule out scenario #1, which means (a, 4) and (a, b+1) define the SAME point. So, we can be certain that b+1 = 4, which means b = 3

Now that we know that b = 3, we can use the fact that the point (4,b) is on the line.
This means that the point (4,3) is on the line y = kx + 1.
When we plug x=4 and y=3 into the equation, we get 3 = (k)(4) + 1
Solve to get k = 1/2

Answer: A

Cheers,
Brent

[vipulgoyal]

As Brent said

(4,b),(a,4),(a,b+1) lie on the same line so they must share same slope

4-b/a-4 = b+1-4/a-a = b+1-b/a-4
4-b/a-4 = 0 = 1/a-4
4-b=1
b=3
(4,b) is on the line.
plug x=4 and y=3 into the equation, we get 3 = (k)(4) + 1
Solve to get k = 1/2