Difference between revisions of "2019 AMC 10B Problems/Problem 4"
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If all lines satisfy the condition, then we can just plug in values for <math>a</math>, <math>b</math>, and <math>c</math> that form an arithmetic progression. Let's use <math>a=1</math>, <math>b=2</math>, <math>c=3</math>, and <math>a=1</math>, <math>b=3</math>, <math>c=5</math>. Then the two lines we get are: <cmath>x+2y=3</cmath> <cmath>x+3y=5</cmath> | If all lines satisfy the condition, then we can just plug in values for <math>a</math>, <math>b</math>, and <math>c</math> that form an arithmetic progression. Let's use <math>a=1</math>, <math>b=2</math>, <math>c=3</math>, and <math>a=1</math>, <math>b=3</math>, <math>c=5</math>. Then the two lines we get are: <cmath>x+2y=3</cmath> <cmath>x+3y=5</cmath> | ||
Use elimination to deduce <cmath>y = 2</cmath> and plug this into one of the previous line equations. We get <cmath>x+4 = 3 \Rightarrow x=-1</cmath> Thus the common point is <math>\boxed{\textbf{(A) } (-1,2)}</math>. | Use elimination to deduce <cmath>y = 2</cmath> and plug this into one of the previous line equations. We get <cmath>x+4 = 3 \Rightarrow x=-1</cmath> Thus the common point is <math>\boxed{\textbf{(A) } (-1,2)}</math>. | ||
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==Solution 2== | ==Solution 2== |
Revision as of 22:17, 17 February 2019
Contents
Problem
All lines with equation such that
form an arithmetic progression pass through a common point. What are the coordinates of that point?
Solution 1
If all lines satisfy the condition, then we can just plug in values for ,
, and
that form an arithmetic progression. Let's use
,
,
, and
,
,
. Then the two lines we get are:
Use elimination to deduce
and plug this into one of the previous line equations. We get
Thus the common point is
.
Solution 2
We know that ,
, and
form an arithmetic progression, so if the common difference is
, we can say
Now we have
, and expanding gives
Factoring gives
. Since this must always be true (regardless of the values of
and
), we must have
and
, so
and the common point is
.
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.