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

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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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New post 11 Jul 2017, 01:33
10
00:00
A
B
C
D
E

Difficulty:

  35% (medium)

Question Stats:

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

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

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New post 11 Jul 2017, 04:29
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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

Answer: C

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

The answer is C
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Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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New post 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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Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va  [#permalink]

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New post 09 Aug 2018, 05:35
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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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