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Two lines, k and m, intersect in the coordinate plane at point (a,b).

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Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   14 Nov 2011, 08:32
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Two lines, k and m, intersect in the coordinate plane at point (a,b). The x-intercept of k and the y-intercept of m are both positive. Is ab positive?

(1) The x-intercept of m and the y-intercept of k are both negative.
(2) The slopes of k and m are both positive.

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Can someone explain the option B graphically.
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D
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sudish
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   14 Nov 2011, 10:43
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Here's the explanation for option B

We are told that both Lines m and k have positive slopes. Any line with a positive slope will slope upwards i.e. will be tilted from left to right. See the figure attached.
We are also told (in the given question) that the x-intercept of k and y-intercept of m are both positive. So, there's only one i.e. unique way to draw these two lines. They are drawn as shown in the attached figure.

Hence, it is quite clear from the diagramming that the point of intersection of these two lines (m and k) considering that they have different slopes, lies in the Third Quadrant, as shown in the figure.

Hence, as 'a' and 'b' both are negative, the product 'ab' is positive. Hence this Statement is sufficient to answer the question alone.
Hope you've already figured out that Statement 1 is also sufficient by itself to answer the question.

Hence the final answer is Option D.

Hope this helps. :)

Cheers!
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   15 Nov 2011, 21:59
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BDSunDevil
@Sudish
Could you please check the the attachment. I have different graph, with same answer, i. e. a and b can either be positive or negative.


Both the diagrams are correct. They are the two different cases that are possible. Let me explain.

The question is: "Is ab positive?"
ab will be positive when either both a and b are positive or both a and b are negative. Both a and b will be positive in the first quadrant and both a and b will be negative in the third quadrant. So if your lines intersect in the first or the third quadrant, ab will be positive.

Each statement alone tells you two things:
1. y intercept of m is positive and x intercept in negative i.e. slope of m is positive
2. y intercept of k is negative and x intercept is positive i.e. slope of k is positive

In this case, k and m cannot intersect in the second or fourth quadrant. They must intersect in the first of third quadrant as shown by the two diagrams you guys have put up.
Hence each statement alone is sufficient.
Answer (D).
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   01 Sep 2014, 03:26
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PathFinder007

HI Karishma,

Could you please explain following statement

y intercept of m is positive and x intercept in negative i.e. slope of m is positive ?

how the slope is coming positive.

Thanks.

What does y intercept positive mean? It means the line intersects the y axis at a point above 0. Similarly, what does x intercept negative mean? It means the line meets the x axis at a point to the left of 0. So try to draw such a line. It will always have positive slope.
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   15 Nov 2011, 10:42
@Sudish
Could you please check the the attachment. I have different graph, with same answer, i. e. a and b can either be positive or negative.
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   04 Jan 2012, 06:07
+1 D.
The two ways in which the lines can be constructed have already been discussed above. It is clear that the point of intersection will lie in either the first or the third quadrant. In either case, a and b will have the same sign and hence, ab will always be positive. Both statements are sufficient to give us that information.
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   06 Jan 2012, 19:09
Re-phrase: (a,b) for lines k amd m will be only positive if that point lies in I or III quadrant. That is k and m must have positive slopes, that is negative intercepts for x-inercept for n=m and y intercept for k respectively.

1. Gives one of the above condition, suff
2. Gives the other condition. suff

D.
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   01 Sep 2014, 01:10
VeritasPrepKarishma
BDSunDevil
@Sudish
Could you please check the the attachment. I have different graph, with same answer, i. e. a and b can either be positive or negative.


Both the diagrams are correct. They are the two different cases that are possible. Let me explain.

The question is: "Is ab positive?"
ab will be positive when either both a and b are positive or both a and b are negative. Both a and b will be positive in the first quadrant and both a and b will be negative in the third quadrant. So if your lines intersect in the first or the third quadrant, ab will be positive.

Each statement alone tells you two things:
1. y intercept of m is positive and x intercept in negative i.e. slope of m is positive
2. y intercept of k is negative and x intercept is positive i.e. slope of k is positive

In this case, k and m cannot intersect in the second or fourth quadrant. They must intersect in the first of third quadrant as shown by the two diagrams you guys have put up.
Hence each statement alone is sufficient.
Answer (D).

HI Karishma,

Could you please explain following statement

y intercept of m is positive and x intercept in negative i.e. slope of m is positive ?

how the slope is coming positive.

Thanks.
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   17 Oct 2020, 04:25
sudish


Thanks a lot! You have the best explanation for the 2nd statement by giving an illustration of the two lines in the X Y coordinate plane.

Thanks again!
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink] Two lines, k and m, intersect in the coordinate plane at point (a,b).   30 May 2023, 05:58
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Re: Two lines, k and m, intersect in the coordinate plane at point (a,b). [#permalink]
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