Two line l and k intersect at a point (4,3). Is the product of their slopes -1
1) x intercepts of line l and k are positive
2) y intercept of line l and k are -ve

please don't write "-ve". take the extra one second to write out the word "negative" in full.
we have readers who don't know what "-ve" is, and will misinterpret it as an algebraic expression.
thank you.

the problem doesn't specify that the lines are non-vertical? hmm.
if this is an official problem, i'm more than a little surprised at the fact that they wouldn't include some little housekeeping fact like that in there.

"product of slopes is -1" should be IMMEDIATELY translated to "perpendicular".

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as with most other coordinate problems, you should try to draw this one first. DO NOT USE MX + B ON COORDINATE PROBLEMS, UNLESS VISUALIZATION FAILS OR THE PROBLEM IS OBVIOUSLY AN ALGEBRA-BASED PROBLEM.

this problem boils down to systematically trying different kinds of lines through the given point and seeing whether you can make them perpendicular (it's obvious that you can make them non-perpendicular).

statement (1)
if both x-intercepts are positive, then the lines could be perpendicular: draw one line ALMOST vertical, but with a positive slope, through (4, 3), so that its x-intercept is slightly less than 4. then the perpendicular will have a very gentle negative slope, and it will have a huge positive x-intercept.
it's easy to make lines with positive x-intercepts, intersecting at the given point, that aren't perpendicular.
insufficient.

statement (2)
any line with a negative y-intercept that also goes through (4, 3) must have a positive slope. since both lines have positive slopes, they can't be perpendicular. (alternatively, it's also easy to use the non-rephrased question here: if both slopes are positive, then the product of the slopes is positive, so it can't be -1.)
sufficient.

ans (b)

Actually the GMAT prep question is phrased as:
In the xy-coordinate plane, like 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 lines l and k is positive.
2) The product of the y-intercepts of lines l and k is negative.

What would be the solution now?

(1) since x-intercepts are both positive. y intercepts could be both could be either positive or negative, while passing through (4,3)
since slope = - ( y intercept / x intercept)
slope can be either positive or negative. So Insufficient

(2) Since line passes through (4,3) and some point on y axis below zero, the slope is positive.
example - (4,3) and (0, -3) slope = (-3-3) / 0 - 4 = 6/4

So B would be my answer.

Ron's having some posting issues, so I'm posting his answer for him:



rephrase: do they have opposite slopes? (i.e., is one of the slopes positive and the other negative?)

statement 1: this means either that both x-intercepts are positive or that both of them are negative.
* if they're both positive, then that doesn't help, since the slopes could either be positive (if the x-intercept is less than 4) or negative (if the x-intercept is greater than 4). INSUFFICIENT.
* (unnecessary to consider at this point, but still) if the x-intercepts are both negative, then both of the slopes must be positive, because the lines must slope upward to get from a negative x-intercept to the point (4, 3).

statement 2: this means that one of the y-intercepts is negative and the other is positive. the line with the negative y-intercept must have a positive slope, because it must slope upward to get from the negative y-intercept to the point (4, 3). the line with the positive intercept could have either a positive slope (if the y-intercept is less than 3) or a negative slope (if the y-intercept is less than 3). this line could also, in fact, have a zero slope, if the y-intercept is exactly 3. INSUFFICIENT.

together: there are exactly two cases to consider, combining the observations from above:
* both x-intercepts are positive, and the y-intercepts are 1 positive 1 negative;
* both x-intercepts are negative, and the y-intercepts are 1 positive 1 negative.

former case: a line with a positive x-intercept and a negative y-intercept must slope upward. a line with a positive x-intercept and a positive y-intercept must slope downward. therefore, the product of the slopes will be negative. SUFFICIENT.

(the latter case is impossible, because one of the lines would have to have a negative x-intercept and a negative y-intercept, a situation that's incompatible with passing through (4, 3).)

_________________
Stacey Koprince
Instructor
Director of Online Community
ManhattanGMAT

For those who hate to visualize and want to see some algebraic expressions..
Line K
y1 = m1x + C1

Line L

y2 = m2X + C2

Is m1m2 < 0

Stmt 1

Product of X intercepts positive ==> -C1/m1 * -C2/m2 = some positive value
Both C1C2 can be positive and hence m1m2 == InSuff

Stmt 2
Product of Yint negative == > C1C2 negative ..
Can't stand alone but fills the logic gap when 2 stmts are taken together
Hence m1m2 must negative

I am having some difficulty understanding why the answer is C. The question asks whether the product of the slope is -1; we have determine that the product of the slopes is negative, but how do we know whether the slope is -1 or another negative number?

The question was written incorrectly: it should read "is the product of their slopes negative?"

Answer should be E.
Line l: y = M1x + C1 passes through (4,3) so 3 = 4M1 + C1. => M1 = (3-C1)/4
Line k: y = M2x + C2 passes through (4,3) so 3 = 4M2 + C2. => M2 = (3-C2)/4

Statement 1: product of x intercepts positive: (C1/M1) * (C2/M2) > 0 ..Don't have C1*C2 product value OR C1,C2 value to find M1*M2. insufficient.

Statement 2: product of y intercepts negative: C1*C2 < 0. Even then cannot evaluate the product of M1 and M2.

Combine 1 and 2. C1*C2 < 0 so M1*M2 < 0. This implies that M1*M2 might be -1 or not. so insufficient.

Hi Ron, so what is the answer of this problem, the GMAT prep one
it should be C ? or ....

yes, (c). note that both statements together are sufficient, while the individual statements are insufficient.

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by the way, please try not to quote text unless you're actually addressing the quoted text.
thanks.

Apology for bumping in the old thread.
I was following the discussion above and good explations.
However , I am finding this whole topic little bit difficult to remember for the actual test. I am of the opinion that this problem will take more than 2 minutes to solve and therefore trying to get important "takeaways" for the posts.
Little bit lost on that front.
Ron / Stacey - would you be able to help by providing some tips ?

Here is what I did.

Slopes should have a negative product. That means they should be in opposite sign.

Statement 1 - Imagine two lines passing through (4,3). Make one line having positive slope and try to rotate the other line around point (4,3). You will see there could be a positive and negative possibility for the other line's slope. So insufficient.

Statement 2 - Again imagine one line to be negative y intercept (slope will be positive). Now - to other line have a negative y intercept it has to have a positive slope. So the product can never be negative.

B should be answer.

hi -- please don't re-post ideas or solutions that are already on the thread.

while correct, these ideas are essentially the basis for my solution posted above:
post23846.html#p23846