Difference between revisions of "2013 AMC 12B Problems/Problem 6"
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==Solution 1== | ==Solution 1== | ||
− | If we move every term dependent on <math>x</math> or <math>y</math> to the LHS, we get <math>x^2 - 10x + y^2 + 6y = -34</math>. Adding <math>34</math> to both sides, we have <math>x^2 - 10x + y^2 + 6y + 34 = 0 \rightarrow x^2-10x+25 + y^2+6y+9 = 0 \rightarrow (x-5)^2+(y+3)^2 = 0</math> Notice this is a circle with radius <math>0</math>, which only contains one point. We can split the <math>34</math> into <math>25</math> and <math>9</math> to get <math>(x - 5)^2 + (y + 3)^2 = 0</math>. So, the only point is <math>(5, -3)</math>, so the sum is <math>5 + (-3) = 2 \implies \boxed{\textbf{(B)}}</math>. ~ asdf334 | + | If we move every term dependent on <math>x</math> or <math>y</math> to the LHS, we get <math>x^2 - 10x + y^2 + 6y = -34</math>. Adding <math>34</math> to both sides, we have <math>x^2 - 10x + y^2 + 6y + 34 = 0 \rightarrow x^2-10x+25 + y^2+6y+9 = 0 \rightarrow (x-5)^2+(y+3)^2 = 0</math>. Notice this is a circle with radius <math>0</math>, which only contains one point. We can split the <math>34</math> into <math>25</math> and <math>9</math> to get <math>(x - 5)^2 + (y + 3)^2 = 0</math>. So, the only point is <math>(5, -3)</math>, so the sum is <math>5 + (-3) = 2 \implies \boxed{\textbf{(B)}}</math>. ~ asdf334 |
==Solution 2== | ==Solution 2== |
Revision as of 18:03, 14 July 2021
- The following problem is from both the 2013 AMC 12B #6 and 2013 AMC 10B #11, so both problems redirect to this page.
Problem
Real numbers and
satisfy the equation
. What is
?
Solution 1
If we move every term dependent on or
to the LHS, we get
. Adding
to both sides, we have
. Notice this is a circle with radius
, which only contains one point. We can split the
into
and
to get
. So, the only point is
, so the sum is
. ~ asdf334
Solution 2
If we move every term including or
to the LHS, we get
We can complete the square to find that this equation becomes
Since the square of any real number is nonnegative, we know that the sum is greater than or equal to
. Equality holds when the value inside the parhentheses is equal to
. We find that
and the sum we are looking for is
- Honestly
Video Solution
https://youtu.be/ba6w1OhXqOQ?t=1810
~ pi_is_3.14
Video Solution
–no one
See Also
2013 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
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 12 Problems and Solutions |
2013 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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.