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If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va
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24 Mar 2021, 11:36
24 Mar 2021, 11:36
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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
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
(A) −1
(B) 0
(C) 1
(D) 2
(E) 3
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
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Posts: 6218
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Re: If the points (a, 0) , (0, b) and (1, 1) are collinear, what is the va
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24 Mar 2021, 12:51
24 Mar 2021, 12:51
1
Carcass 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
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
(A) −1
(B) 0
(C) 1
(D) 2
(E) 3
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
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
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24 Mar 2021, 23:03
24 Mar 2021, 23:03
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Carcass 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
(A) −1
(B) 0
(C) 1
(D) 2
(E) 3
If the points are collinear, they will have sale slope
Apply Slope formula;
\(m = \frac{(b-0)}{(0-a)} = \frac{(1-b)}{(1-0)} = \frac{(1-0)}{(1-a)}\)
\(m = \frac{-b}{a} = \frac{(1-b)}{1} = \frac{1}{(1-a)}\)
Let's pick up this case: \(\frac{-b}{a} = \frac{(1-b)}{1}\)
\(-b = a - ab\)
\(ab = a + b\) .... (1)
Now, we need to find \(\frac{1}{a} + \frac{1}{b}\)
\(\frac{1}{a} + \frac{1}{b} = \frac{(a+b)}{ab}\)
Plug in the value of \(ab\) from eqn (1)
\(\frac{1}{a} + \frac{1}{b} = \frac{(a+b)}{ab} = \frac{(a+b)}{(a+b)} = 1\)
Hence, option C
