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

Question

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  D 
Please explain

hagadre · Answer

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.

Aleehsgonji · Answer

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.

So answer is D

LenaA · Answer

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...

flyingbunny · Answer

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

the answer is D

sesagoa · Answer

Both statements are sufficient alone OA is D

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

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