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Difference between revisions of "2015 AMC 12A Problems/Problem 21"

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==Solution 2 (Basically Solution 1)==
 
==Solution 2 (Basically Solution 1)==
−
To obtain the lower bound, refer to the above solution.
+
As above, we can show that the foci of the ellipse are <math>(\sqrt{15},0), (-\sqrt{15},0).</math>
 +
 
 +
To obtain the lower bound, note that the smallest circle is when the diameter is on the line segment formed by the two foci. We can check that this indeed passes through four points on the ellipse since <math>\sqrt{15}>1.</math> So <math>a=\sqrt{15}.</math>
 +
 
 
To get the upper bound, note that the circle must go through either <math>(0,1)</math> or <math>(0,-1).</math> WLOG, let let the circle go through <math>(0,1).</math> We know that the circle must go through the foci of the ellipse <math>(\sqrt{15},0), (-\sqrt{15},0),</math> So we can apply power of a point to find the diameter. Let <math>x</math> denote the length of the line segment from the origin to the lower point on the circle. Note that <math>x</math> lies on the diameter. Then by POP, we have
 
To get the upper bound, note that the circle must go through either <math>(0,1)</math> or <math>(0,-1).</math> WLOG, let let the circle go through <math>(0,1).</math> We know that the circle must go through the foci of the ellipse <math>(\sqrt{15},0), (-\sqrt{15},0),</math> So we can apply power of a point to find the diameter. Let <math>x</math> denote the length of the line segment from the origin to the lower point on the circle. Note that <math>x</math> lies on the diameter. Then by POP, we have
 
<math>x * 1 = \sqrt{15} * \sqrt{15},</math> yielding <math>x=15</math>, and so the radius of the circle is <math>(15+1)/2=8,</math> so <math>b=8.</math>
 
<math>x * 1 = \sqrt{15} * \sqrt{15},</math> yielding <math>x=15</math>, and so the radius of the circle is <math>(15+1)/2=8,</math> so <math>b=8.</math>
 +
Thus <math>a+b=\sqrt{15}+8.</math>
  
 
~ ccx09 (Roy Short)
 
~ ccx09 (Roy Short)

Revision as of 17:40, 2 December 2017

Problem

A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16.$ The set of all possible values of $r$ is an interval $[a,b).$ What is $a+b?$

$\textbf{(A)}\ 5\sqrt{2}+4\qquad\textbf{(B)}\ \sqrt{17}+7\qquad\textbf{(C)}\ 6\sqrt{2}+3\qquad\textbf{(D)}\ \sqrt{15}+8\qquad\textbf{(E)}\ 12$

Solution

We can graph the ellipse by seeing that the center is $(0, 0)$ and finding the ellipse's intercepts. The points where the ellipse intersects the coordinate axes are $(0, 1), (0, -1), (4, 0)$, and $(-4, 0)$. Recall that the two foci lie on the major axis of the ellipse and are a distance of $c$ away from the center of the ellipse, where $c^2 = a^2 - b^2$, with $a$ being half the length of the major (longer) axis and $b$ being half the minor (shorter) axis of the ellipse. We have that $c^2 = 4^2 - 1^2 \implies$ $c^2 = 15 \implies c = \pm \sqrt{15}$. Hence, the coordinates of both of our foci are $(\sqrt{15}, 0)$ and $(-\sqrt{15}, 0)$. In order for a circle to pass through both of these foci, we must have that the center of this circle lies on the y-axis.

The minimum possible value of $r$ belongs to the circle whose diameter's endpoints are the foci of this ellipse, so $a = \sqrt{15}$. The value for $b$ is achieved when the circle passes through the foci and only three points on the ellipse, which is possible when the circle touches $(0, 1)$ or $(0, -1)$. Which point we use does not change what value of $b$ is attained, so we use $(0, -1)$. Here, we must find the point $(0, y)$ such that the distance from $(0, y)$ to both foci and $(0, -1)$ is the same. Now, we have the two following equations. \[(\sqrt{15})^2 + (y)^2 = b^2\] \[y + 1 = b \implies y = b - 1\] Substituting for $y$, we have that \[15 + (b - 1)^2 = b^2 \implies -2b + 16 = 0.\]

Solving the above simply yields that $b = 8$, so our answer is $a + b = \sqrt{15} + 8 \textbf{ (D)}$.

Solution 2 (Basically Solution 1)

As above, we can show that the foci of the ellipse are $(\sqrt{15},0), (-\sqrt{15},0).$

To obtain the lower bound, note that the smallest circle is when the diameter is on the line segment formed by the two foci. We can check that this indeed passes through four points on the ellipse since $\sqrt{15}>1.$ So $a=\sqrt{15}.$

To get the upper bound, note that the circle must go through either $(0,1)$ or $(0,-1).$ WLOG, let let the circle go through $(0,1).$ We know that the circle must go through the foci of the ellipse $(\sqrt{15},0), (-\sqrt{15},0),$ So we can apply power of a point to find the diameter. Let $x$ denote the length of the line segment from the origin to the lower point on the circle. Note that $x$ lies on the diameter. Then by POP, we have $x * 1 = \sqrt{15} * \sqrt{15},$ yielding $x=15$, and so the radius of the circle is $(15+1)/2=8,$ so $b=8.$ Thus $a+b=\sqrt{15}+8.$

~ ccx09 (Roy Short)

See Also

2015 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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


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