Straight line passes through the points (a,b) and (c,d). Is : DS Archive
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# Straight line passes through the points (a,b) and (c,d). Is

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Manager
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28 Aug 2009, 05:17
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Straight line passes through the points (a,b) and (c,d). Is the slope of the line is less than 0?
(1) (a-c)(b-d)<0
(2) Product of the intercepts of the line on the X-axis and on the Y-axis is greater than 0

OA
[Reveal] Spoiler:
D

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28 Aug 2009, 09:26
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I could prove that the 1st statement is enough, but struggling with secone one.

Basically, the question is asking (d-b)/(c-a)<0
i.e. (d-b) and (c-a) have opposite signs
1) (a-c)(b-d)<0, this is same as (c-a)(d-b) <0. Therefore, they have different signs and hence SUFF

2) Product of x-intercept and y-intercept <0

eqn of line with two pts (x1,y1) & (x2,y2) is y-y1 = m(x-x1) substituting m = ((d-b)/(c-a)), we get

y = mx-mx1+y1, where y1-mx1 is the y-intercept and x-intercept = mx1-y1/m

i tried plugging in the values but not getting the solution.
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28 Aug 2009, 10:02
Two points are (a,b) and (c,d)
1) Slope of the line = (d-b)/(c-a)
Given (d-b)(c-a) <0, so even the slope < 0. Hence statement 1 alone is sufficient.

2) Given product of x intercept and y intercept is greater than 0.
let the points at which the line meets X and Y axis are (x1, 0) and (0, y1)
So slope of line = - y1/x1
Given x1*y1 >0
So slope of the line will be negative. Hence statement 2 alone is sufficient.

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28 Aug 2009, 11:38
Statement 1.
(a-c)(b-d)<0.
1) a-c>0, then b-d<0
2) a-c<0, then b-d>0
It means that when x increases y decreses=> the line is sloping down
or mathematically, the slope $$\frac{a-c}{b-d}<0$$ since numerator and denominator have opposite signs.
SUFFICIENT

Statement 2 is sufficient
If the product of intercepts is positive, intercepts should be of the same sign => this is only the case when a line is sloping down. If the line is sloping "up" the product of the intercepts is negative or zero. Note: if it is said that the product is zero, we can't determine if the line is loping up or down since both are possible.
SUFFICIENT.
Mathematically it can be also shown:
$$y=\alpha\times x +\beta$$
Point (0,$$\beta$$) is the Y intercept
Point $$(- \frac{\beta}{\alpha},0)$$is the X intercept
$$-\frac{\beta}{\alpha}\times \beta=-\frac{\beta^2}{\alpha}>0$$...it is only possible when \alpha<0...
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28 Aug 2009, 21:41
suppose the line is:

y=s*x+k

b=as+k
d=cs+k

s=(b-d)/(a-c)

1) is suff

for 2), just draw the line in the xy plane, it is suff

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29 Aug 2009, 13:24
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Both statements are sufficient alone
OA is D
Re: GmatScore: Line   [#permalink] 29 Aug 2009, 13:24
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