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11 Jul 2017, 02:33
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C
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Difficulty:
45%
(medium)
Question Stats:
66% (02:00) correct
34%
(01:47)
wrong
based on 340
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Not Attempted Yet
If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the value of 1/a + 1/b?
(A) −1
(B) 0
(C) 1
(D) 2
(E) 3
(A) −1
(B) 0
(C) 1
(D) 2
(E) 3
Updated on: 14 Sep 2020, 08:22
Originally posted by BrentGMATPrepNow on 11 Jul 2017, 05:29.
Last edited by BrentGMATPrepNow on 14 Sep 2020, 08:22, edited 1 time in total.
Last edited by BrentGMATPrepNow on 14 Sep 2020, 08:22, edited 1 time in total.
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Bunuel
IMPORTANT CONCEPT: If 3 points are collinear, those points lie on the same line.
So, the slope between any two points on the line will be equal to any other two points on the line.
nguyendinhtuong has already demonstrated this in the above post, by equating the slope between (a,0) and (1,1) with the slope between (0,b) and (1,1)
I just want to follow up on this and show that the strategy works with any 2 pairs of points.
Let's find the slope between (a,0) and (0,b) and the slope between (0,b) and (1,1)
Slope between (a,0) and (0,b) = (b-0)/(0-a) = b/(-a)
Slope between (0,b) and (1,1) = (1-b)/(1-0) = (1-b)/1
So, it must be the case that b/(-a) = (1-b)/1
Cross multiply to get: (1)(b) = (-a)(1-b)
Simplify: b = -a + ab
Add a to both sides: b + a = ab
Divide both sides by ab to get: b/ab + a/ab = ab/ab
Simplify: 1/a + 1/b = 1
Answer: C
General Discussion
11 Jul 2017, 02:55
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Bunuel
The slope of the line that goes through two points (a, 0) and (1, 1) is \(\frac{1-a}{1}\)
The slope of the line that goes through two points (0, b) and (1, 1) is \(\frac{1}{1-b}\)
Since those 3 points are collinear, we have
\(\frac{1-a}{1} = \frac{1}{1-b} \implies (1-a)(1-b)=1 \implies a+b=ab \implies \frac{1}{a} + \frac{1}{b} = 1\)
The answer is C
