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GMATPREP: Coordinate Geometry

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by zaarathelab » Tue Dec 27, 2011 5:37 am
In the XY-coordinate plane, line L and line K intersect at the point (4, 3). Is the product of their slopes negative?
(1) The product of the x-intercepts of line L and K is positive.
(2) The product of the y-intercepts of line L and k is negative.

What is the fastest way to solve this?

I took 6 minutes.
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Source: — Data Sufficiency |

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by Anurag@Gurome » Tue Dec 27, 2011 5:56 am
zaarathelab wrote:In the XY-coordinate plane, line L and line K intersect at the point (4, 3). Is the product of their slopes negative?
(1) The product of the x-intercepts of line L and K is positive.
(2) The product of the y-intercepts of line L and k is negative.

What is the fastest way to solve this?

I took 6 minutes.

Let the equation of line L be y = m1x + c1.
Let the equation of line K be y = m2x + c2.
We need to know whether m1 * m2 is negative or not.

(1) x-intercept of L is -c1/m1.
x-intercept of K is -c2/m2.
So, (c1 * c2)/(m1 * m2) > 0.
But this does not tell us whether m1 * m2 < 0 or not.
So, (1) alone is NOT sufficient.

(2) y-intercept of L is c1.
y-intercept of K is c2.
So, c1 * c2 < 0.
Again this does not tell us whether m1 * m2 < 0 or not.
So, (2) alone is NOT sufficient.

Next, combine both the statements together and check.
On combining we have that (c1 * c2)/(m1 * m2) > 0 and c1 * c2 < 0.
This automatically means that m1 * m2 < 0.
So, both statements together are sufficient to answer the question.

The correct answer is C.
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by [email protected] » Fri Feb 17, 2012 4:40 pm
In the XY-coordinate plane, line L and line K intersect at the point (4, 3). Is the product of their slopes negative?
(1) The product of the x-intercepts of line L and K is positive.
(2) The product of the y-intercepts of line L and k is negative.


A rather simpler one to get...

slope of a line = -b/a

statement 1 alone is not sufficient as there are 3 possibilities of values that can be created...

if 'a' is positive, then the overall slope is negative.

if 'a' is negative, then the overall slope is positive.

also when 'b' is negative and positive, those combinations need to be taken into consideration...


Same with statement 2 alone: not sufficient to answer one specific value...


Combined:


Considering all the possibilities, the final product of the slopes of both the lines comes out to be negative... Let us take all the cases one by one:


in the formula - (b/a), 'a' for both the lines will either be negative or positive.

NOt something like for line 1 'a' is positive and for line 2 'a' is negative (according to statement 1)...


in the formula - (b/a), 'b' for one line will be negative and for the other line it will be positive...

Case 1: 'a' is positive for both the lines and 'b' one positive and one negative:

the overall product of their slopes will turn out to be negative.


Case 2: 'a' is negative for both the lines and 'b' one positive and one negative:

the overall product of their slopes will turn out to be negative.





Hope this helped guyzzz!!!
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