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# If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va

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Math Expert
Joined: 02 Sep 2009
Posts: 51067
If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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11 Jul 2017, 01:33
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Difficulty:

35% (medium)

Question Stats:

70% (02:03) correct 30% (01:56) wrong based on 170 sessions

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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

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Joined: 11 Sep 2015
Posts: 3222
Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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11 Jul 2017, 04:29
3
Top Contributor
4
Bunuel wrote:
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

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

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##### General Discussion
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If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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11 Jul 2017, 01:55
4
Bunuel wrote:
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

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
$$\frac{1-a}{1} = \frac{1}{1-b} \implies (1-a)(1-b)=1 \implies a+b=ab \implies \frac{1}{a} + \frac{1}{b} = 1$$

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Schools: Zicklin '17
Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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23 Jul 2017, 10:51
My approach was formula based and it saved my time:
There are three equations of a line
1. y=mx+c

2. Y-y1/X-x1 = y2-y1/x2-x1

3. X/x intercept +Y/ y intercept =1

for this question I am using 3rd equation
since, (a, 0) , (0, b) are nothing but x and y intercept
so, 1/a + 1/b = 1

Hope it helps. Correct me if needed
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Joined: 09 Sep 2013
Posts: 9088
Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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09 Aug 2018, 05:35
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va &nbs [#permalink] 09 Aug 2018, 05:35
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