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

The OA is C

Source: GMAT Prep

Say the equations of line L and K are

y = m1*x + c1; eqn of line L;
y = m2*x + c2; eqn of line K

Here, m1 and m2 are slopes of line L and line K; and c1 and c2 are y-intercepts of line L and line K.

We have to determine whether m1*m2 < 0.

Let's take each statement one by one.

1) The product of the x-intersects of lines L and K is positive.

Let's transform the equations to get the x-intersects.

We have

y = m1*x + c1=> y - c1 = m1*x => y/m1 - c1/m1 = x => x-intercept of line L = -c1/m1
y = m2*x + c2=> y - c2 = m2*x => y/m2 - c2/m2 = x => x-intercept of line K = -c2/m2

Thus, (-c1/m1)*(-c2/m2) > 0

(c1*c2)/(m1*m2) > 0

Case 1: If c1*c2 > 0, we muat have m1*m2 > 0; the answer is No.
Case 2: If c1*c2 < 0, we must have m1*m2 < 0; the answer is Yes.

No unique answer. Insufficient.

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

=> c1*c2 < 0

It can't help determine whether m1*m2 < 0. Insufficient.

(1) and (2) together

Thus, Case 1 discussed in Statement 1 is invalid; or, only Case 2 is valid; thus, m1*m2 < 0. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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