Difference between revisions of "2022 AIME II Problems/Problem 12"
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Let <math>a, b, x,</math> and <math>y</math> be real numbers with <math>a>4</math> and <math>b>1</math> such that<cmath>\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.</cmath>Find the least possible value of <math>a+b.</math> | Let <math>a, b, x,</math> and <math>y</math> be real numbers with <math>a>4</math> and <math>b>1</math> such that<cmath>\frac{x^2}{a^2}+\frac{y^2}{a^2-16}=\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2}=1.</cmath>Find the least possible value of <math>a+b.</math> | ||
+ | ==Solution== | ||
+ | Denote <math>P = \left( x , y \right)</math>. | ||
+ | |||
+ | Because <math>\frac{x^2}{a^2}+\frac{y^2}{a^2-16} = 1</math>, <math>P</math> is on an ellipse whose center is <math>\left( 0 , 0 \right)</math> and foci are <math>\left( - 4 , 0 \right)</math> and <math>\left( 4 , 0 \right)</math>. | ||
+ | |||
+ | Hence, the sum of distance from <math>P</math> to <math>\left( - 4 , 0 \right)</math> and <math>\left( 4 , 0 \right)</math> is equal to twice the major axis of this ellipse, <math>2a</math>. | ||
+ | |||
+ | Because <math>\frac{(x-20)^2}{b^2-1}+\frac{(y-11)^2}{b^2} = 1</math>, <math>P</math> is on an ellipse whose center is <math>\left( 20 , 11 \right)</math> and foci are <math>\left( 20 , 10 \right)</math> and <math>\left( 20 , 12 \right)</math>. | ||
+ | |||
+ | Hence, the sum of distance from <math>P</math> to <math>\left( 20 , 10 \right)</math> and <math>\left( 20 , 12 \right)</math> is equal to twice the major axis of this ellipse, <math>2b</math>. | ||
+ | |||
+ | Therefore, <math>2a + 2b</math> is the sum of the distance from <math>P</math> to four foci of these two ellipses. | ||
+ | |||
+ | To make this minimized, <math>P</math> is the intersection point of the line that passes through <math>\left( - 4 , 0 \right)</math> and <math>\left( 20 , 10 \right)</math>, and the line that passes through <math>\left( 4 , 0 \right)</math> and <math>\left( 20 , 12 \right)</math>. | ||
+ | |||
+ | The distance between <math>\left( - 4 , 0 \right)</math> and <math>\left( 20 , 10 \right)</math> is <math>\sqrt{\left( 20 + 4 \right)^2 + \left( 10 - 0 \right)^2} = 26</math>. | ||
+ | |||
+ | The distance between <math>\left( 4 , 0 \right)</math> and <math>\left( 20 , 12 \right)</math> is <math>\sqrt{\left( 20 - 4 \right)^2 + \left( 12 - 0 \right)^2} = 20</math>. | ||
+ | |||
+ | Hence, <math>2 a + 2 b = 26 + 20 = 46</math>. | ||
+ | |||
+ | Therefore, <math>a + b = \boxed{\textbf{(023) }}.</math> | ||
+ | |||
+ | ~Steven Chen (www.professorchenedu.com) | ||
==Video Solution== | ==Video Solution== | ||
https://youtu.be/4qiu7GGUGIg | https://youtu.be/4qiu7GGUGIg | ||
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~MathProblemSolvingSkills.com | ~MathProblemSolvingSkills.com | ||
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==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=II|num-b=11|num-a=13}} | {{AIME box|year=2022|n=II|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 12:21, 7 February 2023
Contents
Problem
Let and be real numbers with and such thatFind the least possible value of
Solution
Denote .
Because , is on an ellipse whose center is and foci are and .
Hence, the sum of distance from to and is equal to twice the major axis of this ellipse, .
Because , is on an ellipse whose center is and foci are and .
Hence, the sum of distance from to and is equal to twice the major axis of this ellipse, .
Therefore, is the sum of the distance from to four foci of these two ellipses.
To make this minimized, is the intersection point of the line that passes through and , and the line that passes through and .
The distance between and is .
The distance between and is .
Hence, .
Therefore,
~Steven Chen (www.professorchenedu.com)
Video Solution
~MathProblemSolvingSkills.com
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.