Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1985 AIME Problems/Problem 11"

m (→‎Solution: minor)
(→‎Solution)
Line 3: Line 3:
  
 
== Solution ==
 
== Solution ==
−
An ellipse is defined to be the [[locus]] of points <math>P</math> such that the sum of the distances between <math>P</math> and the two foci is constant.  Let <math>F_1 = (9, 20)</math>, <math>F_2 = (49, 55)</math> and <math>X = (x, 0)</math> be the point of tangency of the ellipse with the <math>x</math>-axis.  Then <math>X</math> must be the point on the axis such that the sum <math>F_1X + F_2X</math> is minimal.  Finding the optimal location for <math>X</math> is a classic problem: for any path from <math>F_1</math> to <math>X</math> and then back to <math>F_2</math>, we can reflect the second leg of this path (from <math>X</math> to <math>F_2</math>) across the <math>x</math>-axis.  Then our path connects <math>F_1</math> to the reflection <math>F_2'</math> of <math>F_2</math> via some point on the <math>x</math>-axis, and this path will have shortest length exactly when our original path has shortest length.  This occurs exactly when we have a straight-line path.   
+
An ellipse is defined to be the [[locus]] of points <math>P</math> such that the sum of the distances between <math>P</math> and the two foci is constant.  Let <math>F_1 = (9, 20)</math>, <math>F_2 = (49, 55)</math> and <math>X = (x, 0)</math> be the point of tangency of the ellipse with the <math>x</math>-axis.  Then <math>X</math> must be the point on the axis such that the sum <math>F_1X + F_2X</math> is minimal.  (The last claim begs justification:  Let <math>F'_2</math> be the reflection of <math>F_2</math> across the <math>y</math>-axis. Let <math>Y</math> be where the line through <math>F_1</math> and <math>F’_2</math> intersects the ellipse.  We will show that <math>X=Y</math>. Note that  <math>X F_2 = X F’_2</math> since <math>X</math> is on the <math>x</math>-axis.  Also, since the entire ellipse is on or above the <math>x</math>-axis and the line through <math>F_2</math> and <math>F’_2</math> is perpendicular to the <math>x</math>-axis, we must have <math>F_2 Y \leq F’_2 Y</math> with equality if and only if <math>Y</math> is on the <math>x</math>-axis.  Now, we have
 +
<cmath>F_1 X + F'_2 X = F_1 X + F_2 X = F_1 Y + F_2 Y \leq F_1 Y + F’_2 Y</cmath>
 +
But the right most sum is the straight-line distance from <math> F_1</math> to <math>F’_2</math> and the left is the distance of some path from <math> F_1</math> to <math>F_2</math>., so this is only possible if we have equality and thus <math>X = Y</math>). Finding the optimal location for <math>X</math> is a classic problem: for any path from <math>F_1</math> to <math>X</math> and then back to <math>F_2</math>, we can reflect (as above) the second leg of this path (from <math>X</math> to <math>F_2</math>) across the <math>x</math>-axis.  Then our path connects <math>F_1</math> to the reflection <math>F_2'</math> of <math>F_2</math> via some point on the <math>x</math>-axis, and this path will have shortest length exactly when our original path has shortest length.  This occurs exactly when we have a straight-line path, and by the above argument, this path passes through the point of tangency of the ellipse with the <math>x-</math>axis.   
 
<center><asy>
 
<center><asy>
 
size(200);
 
size(200);

Revision as of 15:49, 6 June 2018

Problem

An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?

Solution

An ellipse is defined to be the locus of points $P$ such that the sum of the distances between $P$ and the two foci is constant. Let $F_1 = (9, 20)$, $F_2 = (49, 55)$ and $X = (x, 0)$ be the point of tangency of the ellipse with the $x$-axis. Then $X$ must be the point on the axis such that the sum $F_1X + F_2X$ is minimal. (The last claim begs justification: Let $F'_2$ be the reflection of $F_2$ across the $y$-axis. Let $Y$ be where the line through $F_1$ and $F’_2$ intersects the ellipse. We will show that $X=Y$. Note that $X F_2 = X F’_2$ since $X$ is on the $x$-axis. Also, since the entire ellipse is on or above the $x$-axis and the line through $F_2$ and $F’_2$ is perpendicular to the $x$-axis, we must have $F_2 Y \leq F’_2 Y$ with equality if and only if $Y$ is on the $x$-axis. Now, we have \[F_1 X + F'_2 X = F_1 X + F_2 X = F_1 Y + F_2 Y \leq F_1 Y + F’_2 Y\] But the right most sum is the straight-line distance from $F_1$ to $F’_2$ and the left is the distance of some path from $F_1$ to $F_2$., so this is only possible if we have equality and thus $X = Y$). Finding the optimal location for $X$ is a classic problem: for any path from $F_1$ to $X$ and then back to $F_2$, we can reflect (as above) the second leg of this path (from $X$ to $F_2$) across the $x$-axis. Then our path connects $F_1$ to the reflection $F_2'$ of $F_2$ via some point on the $x$-axis, and this path will have shortest length exactly when our original path has shortest length. This occurs exactly when we have a straight-line path, and by the above argument, this path passes through the point of tangency of the ellipse with the $x-$axis.

[asy] size(200); pointpen=black;pathpen=black+linewidth(0.6);pen f = fontsize(10); pair F1=(9,20),F2=(49,55); D(shift((F1+F2)/2)*rotate(41.186)*scale(85/2,10*11^.5)*unitcircle); D((-20,0)--(80,0)--(0,0)--(0,80)--(0,-60)); path p = F1--(49,-55); pair X = IP(p,(0,0)--(80,0)); D(p,dashed);D(F1--X--F2);D(F1);D(F2);D((49,-55)); MP("X",X,SW,f); MP("F_1",F1,NW,f); MP("F_2",F2,NW,f); MP("F_2'",(49,-55),NE,f); [/asy]

The sum of the two distances $F_1 X$ and $F_2X$ is therefore equal to the length of the segment $F_1F_2'$, which by the distance formula is just $d = \sqrt{(9 - 49)^2 + (20 + 55)^2} = \sqrt{40^2 + 75^2} = 5\sqrt{8^2 + 15^2} = 5\cdot 17 = 85$.

Finally, let $A$ and $B$ be the two endpoints of the major axis of the ellipse. Then by symmetry $AF_1 = F_2B$ so $AB = AF_1 + F_1B = F_2B + F_1B = d$ (because $B$ is on the ellipse), so the answer is $\boxed{085}$.

See also

1985 AIME (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
All AIME Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1985_AIME_Problems/Problem_11&oldid=94946"