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If the three points A, B and C with the coordinates (4, 2), (3, 1) and

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If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 30 Jul 2015, 10:20
2
2
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A
B
C
D
E

Difficulty:

  5% (low)

Question Stats:

88% (00:36) correct 12% (01:41) wrong based on 144 sessions

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If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 30 Jul 2015, 10:32
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Bunuel wrote:
If the three points A, B and C with the coordinates (4, 2), (3, 1) and (2, a) respectively lie on a line, what is the value of a?

A. 0
B. 1
C. 2
D. 3
E. 4


The Line will have the same slope irrespective of which two points are taken into account

i.e. Slope of Line using Points (4, 2), (3, 1) = Slope of Line using Points (4, 2), (2, a)

CONCEPT: Slope between two points (a, b) and (c, d) = (d-b)/(c-a)

i.e. (1-2)/(3-4) = (a-2)/(2-4)

i.e. (a-2) = -2

i.e. a = 0

Answer: option A
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If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 30 Jul 2015, 13:25
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Bunuel wrote:
If the three points A, B and C with the coordinates (4, 2), (3, 1) and (2, a) respectively lie on a line, what is the value of a?


Little to add here. To solve, either plot out on a graph or calculate the equation of the line, and in turn, solve for \(a\).
\(y=mx+b\)
\(m = \frac{(2-1)}{(4-3)} = 1\)
\(y = x + b\)
\(2 = 4 + b\)
\(b = -2\)
if \(y = x - 2\), then \(a = 2 - 2\), and therefore \(a = 0\)

IMO A.

Since the equation is the first degree, the line is linear, and thus it cannot have the same y value as the other points. So, B or C are out as possible answers automatically. Plotting the others shows that C and D are wrong since the equation is not a quadratic.
A. 0
B. 1 Out
C. 2 Out
D. 3 Out
E. 4 Out
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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 30 Jul 2015, 13:59
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Hi All,

Since the 3 points have consecutive X-co-ordinate values, we can use 'brute force' to get to the solution without having to do too much math.

The points are...

(4,2)
(3,1)
(2,A)

Looking at the first two co-ordinates, the X-coordinate 'drops' by 1 and the Y-co-ordinate 'drops' by 1. Since we're told that all 3 points lie on the same line, that 'pattern' will continue. Thus, going from the second co-ordinate to the third co-ordinate, since the X-coordinate 'drops' by 1, the Y-co-ordinate MUST 'drop' by 1....

1-1 = 0

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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 30 Jul 2015, 14:20
1
Bunuel wrote:
If the three points A, B and C with the coordinates (4, 2), (3, 1) and (2, a) respectively lie on a line, what is the value of a?

A. 0
B. 1
C. 2
D. 3
E. 4


IMO: A

If the points collinear, then their slopes must be equal.

Slope between two points (x1,y1) (x2,y2) is given by =\(\frac{y2-y1}{x2-x1}\)

Thus Slope AB = Slope BC
\(\frac{1-2}{3-4}\) = \(\frac{a-1}{2-3}\)
a= 0
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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 31 Jul 2015, 00:58
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slope of line = (4-3)/(2-1) = 1
y = mx + c
Putting (3,1) in it we get equation of line to be:- y = x-2
thus, when x = 2, y = a = 0.

Ans.= A
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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 31 Jul 2015, 23:37
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If I simply plot all the points in the xy plane, then it can be noticed that when x-coordinate is 2, then y coordinate must be 0 or negative.
As answer choices don't include any negative y-coordinate, the answer must be 0.
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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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New post 14 Jul 2016, 08:37
Bunuel wrote:
If the three points A, B and C with the coordinates (4, 2), (3, 1) and (2, a) respectively lie on a line, what is the value of a?

A. 0
B. 1
C. 2
D. 3
E. 4


The first two points tells us that the slope = 2-1/1-3 = 1

Meaning all the point has have a constant distance between them

Plotting this info in a table we get
X Y
4 2
3 1
2 0

Answer is A
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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and  [#permalink]

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Re: If the three points A, B and C with the coordinates (4, 2), (3, 1) and &nbs [#permalink] 28 Nov 2018, 14:37
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