Co-Ordinate Geometry

[kamalakarthi]

Hi,

Can you help me with the below question.

[[email protected]]

Hi kamalakarthi,

Based on the drawing - and the fact that the answer choices are relatively "spread out", you can answer this question 'visually.' We're looking for the X-coordinate values on the line in which Y < 0 (meaning that the Y-coordinate is below the X-axis). That certainly appears to occur somewhere between when X=0 and X=4. There's only one answer that logically matches....

Mathematically, we can prove the correct answer. Two points that we know for certain are on the line are (4, 2) and (-2, -4). We can use those two points to find the Slope of the line:

(Change in Y)/(Change in X) = [2 - (-4)]/[4 - (-2)] = 6/6 = 1. With a slope of 1, we can work "up" or "down" from either of those two co-ordinates. If you word "down" from (4,2), you would hit (3,1), (2,0), (1, -1), (0,-2), etc. Thus, when X < 2, we'll have a Y that is less than 0.

Final Answer: D

GMAT assassins aren't born, they're made,
Rich

[Matt@VeritasPrep]

Another trick: notice where the line intercepts the x-axis, then work from there.

We know (2, 0) is on the line and on the x-axis. We know that for all x-values < 2, the line is below the x-axis, and that for all x-values > 2, the line is above the x-axis.

"Below the x-axis" is another way of saying y < 0, so for all x < 2, we'll have y-values on the line < 0.

[kamalakarthi]

Thanks Matt. We know -2 is on the X-axis but how can we 2 is also on the X -axis. What if 1.8 is on the X-axis. This is the part I am having trouble with.

Can you help on what am I missing.

[[email protected]]

Hi kamalakarthi,

You can figure out the equation for the line by determining the Slope and then using the Slope-Intercept Formula. With that Formula, you can then determine every point on that line...

Two points that we know for certain are on the line are (4, 2) and (-2, -4). We can use those two points to find the Slope of the line:

(Change in Y)/(Change in X) = [2 - (-4)]/[4 - (-2)] = 6/6 = 1. With a slope of 1, we can now use the Slope-Intercept Formula:

Y = (M)(X) + B
Y = (1)(X) + B

Using either of the two co-ordinates that we started with, we can figure out the value of B...

(4, 2)
2 = (1)(4) + B
2 = 4 + B
-2 = B

Thus, the equation of the line is:
Y = X - 2

And we have proof that (2,0) is the co-ordinate that crosses the X-axis.

GMAT assassins aren't born, they're made,
Rich

[Matt@VeritasPrep]

Thanks Matt. We know -2 is on the X-axis but how can we 2 is also on the X -axis. What if 1.8 is on the X-axis. This is the part I am having trouble with.

Can you help on what am I missing.

Sure! It can't be on the x-axis, since the lines will only intersect at one point, and we've already got an intersection between the two: x = 2 and y = 0 (the x-axis). Two straight lines can only intersect at one point - this is one of the cornerstones of Euclidean geometry. (Well, they can also intersect at zero points or all points, if they're parallel or identical!)