Difference between revisions of "2024 AIME I Problems/Problem 9"
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==Problem== | ==Problem== | ||
− | Let <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> be | + | Let <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> be points on the hyperbola <math>\frac{x^2}{20}- \frac{y^2}{24} = 1</math> such that <math>ABCD</math> is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than <math>BD^2</math> for all such rhombi. |
− | ==Solution== | + | ==Solution 1== |
A quadrilateral is a rhombus if and only if its two diagonals bisect each other and are perpendicular to each other. The first condition is automatically satisfied because of the hyperbola's symmetry about the origin. To satisfy the second condition, we set <math>BD</math> as the line <math>y = mx</math> and <math>AC</math> as <math>y = -\frac{1}{m}x.</math> Because the hyperbola has asymptotes of slopes <math>\pm \frac{\sqrt6}{\sqrt5},</math> we have <math>m, -\frac{1}{m} \in \left(-\frac{\sqrt6}{\sqrt5}, \frac{\sqrt6}{\sqrt5}\right).</math> This gives us <math>m^2 \in \left(\frac{5}{6}, \frac{6}{5}\right).</math> | A quadrilateral is a rhombus if and only if its two diagonals bisect each other and are perpendicular to each other. The first condition is automatically satisfied because of the hyperbola's symmetry about the origin. To satisfy the second condition, we set <math>BD</math> as the line <math>y = mx</math> and <math>AC</math> as <math>y = -\frac{1}{m}x.</math> Because the hyperbola has asymptotes of slopes <math>\pm \frac{\sqrt6}{\sqrt5},</math> we have <math>m, -\frac{1}{m} \in \left(-\frac{\sqrt6}{\sqrt5}, \frac{\sqrt6}{\sqrt5}\right).</math> This gives us <math>m^2 \in \left(\frac{5}{6}, \frac{6}{5}\right).</math> | ||
− | Plugging <math>y = mx</math> into the equation for the hyperbola yields <math>x^2 = \frac{120}{6-5m^2}</math> and <math>y^2 = \frac{120m^2}{6-5m^2}.</math> By symmetry of the hyperbola, we know that <math>\left(\frac{BD}{2}\right)^2 = x^2 + y^2,</math> so we wish to find a lower bound for <math>x^2 + y^2 = 120\left(\frac{1+m^2}{6-5m^2}\right).</math> This is equivalent to minimizing <math>\frac{1+m^2}{6-5m^2} = -\frac{1}{5} + \frac{11}{5(6-5m^2)}</math>. It's then easy to see that this expression increases with <math>m^2,</math> so we plug in <math>m^2 = \frac{5}{6}</math> to get <math>x^2+y^2 > 120,</math> giving <math>BD^2 > \boxed{480}.</math> | + | Plugging <math>y = mx</math> into the equation for the hyperbola yields <math>x^2 = \frac{120}{6-5m^2}</math> and <math>y^2 = \frac{120m^2}{6-5m^2}.</math> By symmetry of the hyperbola, we know that <math>\left(\frac{BD}{2}\right)^2 = x^2 + y^2,</math> so we wish to find a lower bound for <math>x^2 + y^2 = 120\left(\frac{1+m^2}{6-5m^2}\right).</math> This is equivalent to minimizing <math>\frac{1+m^2}{6-5m^2} = -\frac{1}{5} + \frac{11}{5(6-5m^2)}</math>. It's then easy to see that this expression increases with <math>m^2,</math> so we plug in <math>m^2 = \frac{5}{6}</math> to get <math>x^2+y^2 > 120,</math> giving <math>BD^2 > \boxed{480}.</math> |
==Solution 2== | ==Solution 2== | ||
− | Assume <math>AC</math> is the | + | Assume that <math>AC</math> is the asymptote of the hyperbola, in which case <math>BD</math> is minimized. The expression of <math>BD</math> is <math>y=-\sqrt{\frac{5}{6}}x</math>. Thus, we could get <math>\frac{x^2}{20}-\frac{y^2}{24}=1\implies x^2=\frac{720}{11}</math>. The desired value is <math>4\cdot \frac{11}{6}x^2=480</math>. This case can't be achieved, so all <math>BD^2</math> would be greater than <math>\boxed{480}</math> |
~Bluesoul | ~Bluesoul | ||
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− | + | ==Video Solution== | |
+ | https://youtu.be/9Fxz50ZMk1E?si=O2y5t0VXAAfPPTbv | ||
− | + | ~MathProblemSolvingSkills.com | |
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− | ==Solution | + | ==Video Solution== |
− | + | by OmegaLearn.org https://youtu.be/Ex-IGnoAS48 | |
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− | https://youtu.be/Ex-IGnoAS48 | ||
==Video Solution== | ==Video Solution== |
Latest revision as of 13:01, 24 December 2024
Contents
Problem
Let , , , and be points on the hyperbola such that is a rhombus whose diagonals intersect at the origin. Find the greatest real number that is less than for all such rhombi.
Solution 1
A quadrilateral is a rhombus if and only if its two diagonals bisect each other and are perpendicular to each other. The first condition is automatically satisfied because of the hyperbola's symmetry about the origin. To satisfy the second condition, we set as the line and as Because the hyperbola has asymptotes of slopes we have This gives us
Plugging into the equation for the hyperbola yields and By symmetry of the hyperbola, we know that so we wish to find a lower bound for This is equivalent to minimizing . It's then easy to see that this expression increases with so we plug in to get giving
Solution 2
Assume that is the asymptote of the hyperbola, in which case is minimized. The expression of is . Thus, we could get . The desired value is . This case can't be achieved, so all would be greater than
~Bluesoul
Video Solution
https://youtu.be/9Fxz50ZMk1E?si=O2y5t0VXAAfPPTbv
~MathProblemSolvingSkills.com
Video Solution
by OmegaLearn.org https://youtu.be/Ex-IGnoAS48
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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