Data Sufficiency

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-intersects of lines L and K is positive.
(2) The product of the y-intersects of lines L and K is negative.


Source - Tough 700 Questionnaire.

OA - C


I searched this question on this forum and found that

L = a1x + b1
K = a2x + b2

Slope L : -b1/a1
Slope K : -b2/a2

But the slope form of the equation is y = mx + c, so the above equation should have slope as a1 only and not -b1/a1


If the equation is in standard form Ax + By + C = 0 then y = -A/Bx - C/A
then the slope is -A/b and y-intercept = - C/A


So L = a1x + b1 is in which form?

Lets us write the equation of line in intercept form.

equation of line L is x/a + y/b = 1
equation of line K is x/c + y/d = 1


slope of line L is =-(b/a)
slope of line K is =-(d/c)

product of slopes = -(b/a) * -(d/c)= bd/ac

we have dertermine whether bd/ac is negative or not.


statment1 = a*c >0

from above statment it is not possible to say that bd/ac is nagative. it will depends on bd, if bd is positive then bd/ac is positive and vice versa.

so S1 is not sufficint to decide the answer.

similarlly , SM2 is not sufficient to answer the question clearly. if bd is negative then it is not possible to say that bd/ac is nagative. it will depends on ac, if ac is positive then bd/ac is positive and vice versa.

if both statment consider jointly then it can be answered clearly.

SM1 ac>0
SM2 bd<0

so product of slope i.e. bd/ac <0

so answer is c

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.

Don't solve. Draw.

Statement 1: The product of the x-intercepts of line L and K is positive.
K has a negative x-intercept, L has a negative x-intercept, the product of the slopes is positive:


K has a positive x-intercept, L has a positive x-intercept, the product of the slopes is negative:

INSUFFICIENT.

Statement 2: The product of the y-intercepts of line L and k is negative.
K has a negative y-intercept, L has a positive y-intercept, the product of the slopes is negative:


K has a negative y-intercept, L has a positive y-intercept, the product of the slopes is positive:

INSUFFICIENT.

Statements 1 and 2 combined:
Statement 1 requires that both x-intercepts be negative or that both x-intercepts be positive (so that their product is positive).
If both x-intercepts are negative, then both y-intercepts must be positive, which does not satisfy statement 2:


Thus, both x-intercepts must be positive.
Statement 2 requires that one of the y-intercepts be positive, the other negative.
A positive x-intercept and a positive y-intercept yields a negative slope.
A positive x-intercept and a negative y-intercept yields a positive slope.
See below:

Thus, the product of the slopes must be negative.
SUFFICIENT.

The correct answer is C.

Thank you very much Mitch.
Can you please confirm my below understanding if it is right or wrong?

1. If a line in a plane is directed in a north-east direction it will have a positive slope.
2. if a line in a plane is directed in a south-east direction it will have a negative slope.
3. If a line is parallel to y-axis the slope is not defined.
4. If a line is parallel to x-axis the slope is zero.


So in this way, we have evaluated the four cases in the statement 1 and statement 2 will not poise any problem correct?