Difference between revisions of "2008 AIME I Problems/Problem 14"

Latest revision as of 12:29, 3 September 2024

Problem

Let $\overline{AB}$ be a diameter of circle $\omega$ . Extend $\overline{AB}$ through $A$ to $C$ . Point $T$ lies on $\omega$ so that line $CT$ is tangent to $\omega$ . Point $P$ is the foot of the perpendicular from $A$ to line $CT$ . Suppose $\overline{AB} = 18$ , and let $m$ denote the maximum possible length of segment $BP$ . Find $m^{2}$ .

Solution

Solution 1

$[asy] size(250); defaultpen(0.70 + fontsize(10)); import olympiad; pair O = (0,0), B = O - (9,0), A= O + (9,0), C=A+(18,0), T = 9 * expi(-1.2309594), P = foot(A,C,T); draw(Circle(O,9)); draw(B--C--T--O); draw(A--P); dot(A); dot(B); dot(C); dot(O); dot(T); dot(P); draw(rightanglemark(O,T,C,30)); draw(rightanglemark(A,P,C,30)); draw(anglemark(B,A,P,35)); draw(B--P, blue); label("$A$",A,NW); label("$B$",B,NW); label("$C$",C,NW); label("$O$",O,NW); label("$P$",P,SE); label("$T$",T,SE); label("$9$",(O+A)/2,N); label("$9$",(O+B)/2,N); label("$x-9$",(C+A)/2,N); [/asy]$

Let $x = OC$ . Since $OT, AP \perp TC$ , it follows easily that $\triangle APC \sim \triangle OTC$ . Thus $\frac{AP}{OT} = \frac{CA}{CO} \Longrightarrow AP = \frac{9(x-9)}{x}$ . By the Law of Cosines on $\triangle BAP$ , $\begin{align*}BP^2 = AB^2 + AP^2 - 2 \cdot AB \cdot AP \cdot \cos \angle BAP \end{align*}$ where $\cos \angle BAP = \cos (180 - \angle TOA) = - \frac{OT}{OC} = - \frac{9}{x}$ , so: $\begin{align*}BP^2 &= 18^2 + \frac{9^2(x-9)^2}{x^2} + 2(18) \cdot \frac{9(x-9)}{x} \cdot \frac 9x = 405 + 729\left(\frac{2x - 27}{x^2}\right)\end{align*}$ Let $k = \frac{2x-27}{x^2} \Longrightarrow kx^2 - 2x + 27 = 0$ ; this is a quadratic, and its discriminant must be nonnegative: $(-2)^2 - 4(k)(27) \ge 0 \Longleftrightarrow k \le \frac{1}{27}$ . Thus, $BP^2 \le 405 + 729 \cdot \frac{1}{27} = \boxed{432}$ Equality holds when $x = 27$ .~Shen Kislay Kai

Solution 1.1 (Calculus)

Proceed as follows for Solution 1.

Once you approach the function $k=(2x-27)/x^2$ , find the maximum value by setting $dk/dx=0$ .

Simplifying $k$ to take the derivative, we have $2/x-27/x^2$ , so $dk/dx=-2/x^2+54/x^3$ . Setting $dk/dx=0$ , we have $2/x^2=54/x^3$ .

Solving, we obtain $x=27$ as the critical value. Hence, $k$ has the maximum value of $(2*27-27)/27^2=1/27$ . Since $BP^2=405+729k$ , the maximum value of $\overline {BP}$ occurs at $k=1/27$ , so $BP^2$ has a maximum value of $405+729/27=\fbox{432}$ .

Note: Please edit this solution if it feels inadequate. ~Shen Kislay Kai

Solution 2

$[asy] unitsize(3mm); pair B=(0,13.5), C=(23.383,0); pair O=(7.794, 9), P=(2*7.794,0); pair T=(7.794,0), Q=(0,0); pair A=(2*7.794,4.5); draw(Q--B--C--Q); draw(O--T); draw(A--P); draw(Circle(O,9)); dot(A);dot(B);dot(C);dot(T);dot(P);dot(O);dot(Q); label("$B$",B,NW); label("$A$",A,NE); label("$O$",O,N); label("$P$",P,S); label("$T$",T,S); label("$Q$",Q,S); label("$C$",C,E); label("$\theta$",C + (-1.7,-0.2), NW); label("$9$", (B+O)/2, N); label("$9$", (O+A)/2, N); label("$9$", (O+T)/2,W); [/asy]$

From the diagram, we see that $BQ = OT + BO \sin\theta = 9 + 9\sin\theta = 9(1 + \sin\theta)$ , and that $QP = BA\cos\theta = 18\cos\theta$ .

$\begin{align*}BP^2 &= BQ^2 + QP^2 = 9^2(1 + \sin\theta)^2 + 18^2\cos^2\theta\\ &= 9^2[1 + 2\sin\theta + \sin^2\theta + 4(1 - \sin^2\theta)]\\ BP^2 &= 9^2[5 + 2\sin\theta - 3\sin^2\theta]\end{align*}$

This is a quadratic equation, maximized when $\sin\theta = \frac { - 2}{ - 6} = \frac {1}{3}$ . Thus, $m^2 = 9^2[5 + \frac {2}{3} - \frac {1}{3}] = \boxed{432}$ .

Solution 3 (Calculus Bash)

$[asy] unitsize(3mm); pair B=(0,13.5), C=(23.383,0); pair O=(7.794, 9), P=(2*7.794,0); pair T=(7.794,0), Q=(0,0); pair A=(2*7.794,4.5); draw(Q--B--C--Q); draw(O--T); draw(A--P); draw(Circle(O,9)); dot(A);dot(B);dot(C);dot(T);dot(P);dot(O);dot(Q); label("$B$",B,NW); label("$A$",A,NE); label("$\omega$",O,N); label("$P$",P,S); label("$T$",T,S); label("$Q$",Q,S); label("$C$",C,E); label("$9$", (B+O)/2, N); label("$9$", (O+A)/2, N); label("$9$", (O+T)/2,W); [/asy]$

(Diagram credit goes to Solution 2)

We let $AC=x$ . From similar triangles, we have that $PC=\frac{x\sqrt{x^2+18x}}{x+9}$ (Use Pythagorean on $\triangle\omega TC$ and then using $\triangle\omega CT\sim\triangle ACP$ ). Similarly, $TP=QT=\frac{9\sqrt{x^2+18x}}{x+9}$ . Using the Pythagorean Theorem again and $\triangle CAP\sim\triangle CBQ$ , $BQ=\sqrt{(x+18)^2-(\frac{(x+18)\sqrt{x^2+18x}}{x+9})^2}$ . Using the Pythagorean Theorem $\bold{again}$ , $BP=\sqrt{(x+18)^2-(\frac{(x+18)\sqrt{x^2+18x}}{x+9})^2+(\frac{18\sqrt{x^2+18x}}{x+9})^2}$ . After a large bashful simplification, $BP=\sqrt{405+\frac{1458x-6561}{x^2+18x+81}}$ . The fraction is equivalent to $729\frac{2x-9}{(x+9)^2}$ . Taking the derivative of the fraction and solving for x, we get that $x=18$ . Plugging $x=18$ back into the expression for $BP$ yields $\sqrt{432}$ , so the answer is $(\sqrt{432})^2=\boxed{432}$ .

Solution 4

$[asy] unitsize(3mm); pair B=(0,13.5), C=(23.383,0); pair O=(7.794, 9), P=(2*7.794,0); pair T=(7.794,0), Q=(0,0); pair A=(2*7.794,4.5); draw(Q--B--C--Q); draw(O--T); draw(A--P); draw(Circle(O,9)); dot(A);dot(B);dot(C);dot(T);dot(P);dot(O);dot(Q); label("$B$",B,NW); label("$A$",A,NE); label("$\omega$",O,N); label("$P$",P,S); label("$T$",T,S); label("$Q$",Q,S); label("$C$",C,E); label("$9$", (B+O)/2, N); label("$9$", (O+A)/2, N); label("$9$", (O+T)/2,W); [/asy]$

(Diagram credit goes to Solution 2)

Let $AC=x$ . The only constraint on $x$ is that it must be greater than $0$ . Using similar triangles, we can deduce that $PA=\frac{9x}{x+9}$ . Now, apply law of cosines on $\triangle PAB$ . $BP^2=\left(\frac{9x^2}{x+9}\right)^2+18^2-2(18)\left(\frac{9x}{x+9}\right)\cos(\angle PAB).$ We can see that $\cos(\angle PAB)=\cos(180^{\circ}-\angle PAC)=\cos(\angle PAC -90^{\circ})=-\sin(\angle PCA)$ . We can find $-\sin(\angle PCA)=-\frac{9}{x+9}$ . Plugging this into our equation, we get: $BP^2=\left(\frac{9x^2}{x+9}\right)^2+18^2-2(18)\left(\frac{9x}{x+9}\right)\left(-\frac{9}{x+9}\right).$ Eventually, $BP^2 = 81\left(\frac{x^2+36x}{(x+9)^2}+4\right).$ We want to maximize $\frac{x^2+36x}{(x+9)^2}$ . There are many ways to maximize this expression, discussed here: https://artofproblemsolving.com/community/c4h2292700_maximization. The maximum result of that expression is $\frac{4}{3}$ . Finally, evaluating $BP^2$ for this value $81\left(\frac{4}{3}+4\right) = \boxed{432}$ .

~superagh

Solution 5 (Clean)

Let $h$ be the distance from $A$ to $CT$ . Observe that $h$ takes any value from $0$ to $2r$ , where $r$ is the radius of the circle.

Let $Q$ be the foot of the altitude from $B$ to $CT$ . It is clear that $T$ is the midpoint of $PQ$ , and so the length $OT$ is the average of $AP$ and $BQ$ . It follows thus that $BQ = 2r - h$ .

We compute $PT = \sqrt{r^2 - (r - h)^2} = \sqrt{h(2r - h)},$ and so $BP^2 = PQ^2 + BQ^2 = 4PT^2 + BQ^2 = 4h(2r - h) + (2r-h)^2 = (2r-h)(2r + 3h)$ . This is $\frac{1}{3}(6r - 3h)(2r + 3h) \le \frac{1}{3} \cdot \left( \frac{8r}{2} \right)^2$ . Equality is attained, so thus we extract the answer of $\frac{16 \cdot 9^2}{3} = 27 \cdot 16 = \boxed{432}.$

@@ Line 1: / Line 1: @@
+__TOC__
 == Problem ==
-Let <math>\overline{AB}</math> be a diameter of circle <math>\omega</math>. Extend <math>\overline{AB}</math> through <math>A</math> to <math>C</math>. Point <math>T</math> lies on <math>\omega</math> so that line <math>CT</math> is tangent to <math>\omega</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>AB = 18</math>, and let <math>m</math> denote the maximum possible length of segment <math>BP</math>. Find <math>m^{2}</math>.
+Let <math>\overline{AB}</math> be a diameter of circle <math>\omega</math>. Extend <math>\overline{AB}</math> through <math>A</math> to <math>C</math>. Point <math>T</math> lies on <math>\omega</math> so that line <math>CT</math> is tangent to <math>\omega</math>. Point <math>P</math> is the foot of the perpendicular from <math>A</math> to line <math>CT</math>. Suppose <math>\overline{AB} = 18</math>, and let <math>m</math> denote the maximum possible length of segment <math>BP</math>. Find <math>m^{2}</math>.
 == Solution ==
+=== Solution 1 ===
 <center><asy>
 size(250); defaultpen(0.70 + fontsize(10)); import olympiad;
@@ Line 24: / Line 27: @@
 where <math>\cos \angle BAP = \cos (180 - \angle TOA) = - \frac{OT}{OC} = - \frac{9}{x}</math>, so:
 <cmath>\begin{align*}BP^2 &= 18^2 + \frac{9^2(x-9)^2}{x^2} + 2(18) \cdot \frac{9(x-9)}{x} \cdot \frac 9x = 405 + 729\left(\frac{2x - 27}{x^2}\right)\end{align*}</cmath>
-Let <math>m = \frac{2x-27}{x^2} \Longrightarrow mx^2 - 2x + 27 = 0</math>; this is a quadratic, and its [[discriminant]] must be nonnegative: <math>(-2)^2 - 4(m)(27) \ge 0 \Longleftrightarrow m \le \frac{1}{27}</math>. Thus,
+Let <math>k = \frac{2x-27}{x^2} \Longrightarrow kx^2 - 2x + 27 = 0</math>; this is a quadratic, and its [[discriminant]] must be nonnegative: <math>(-2)^2 - 4(k)(27) \ge 0 \Longleftrightarrow k \le \frac{1}{27}</math>. Thus,
 <cmath>BP^2 \le 405 + 729 \cdot \frac{1}{27} = \boxed{432}</cmath>
-Equality holds when <math>x = 27</math>.
+Equality holds when <math>x = 27</math>.~Shen Kislay Kai
+==== Solution 1.1 (Calculus) ====
+Proceed as follows for Solution 1.
+Once you approach the function <math>k=(2x-27)/x^2</math>, find the maximum value by setting <math>dk/dx=0</math>.
+Simplifying <math>k</math> to take the derivative, we have <math>2/x-27/x^2</math>, so <math>dk/dx=-2/x^2+54/x^3</math>.  Setting <math>dk/dx=0</math>, we have <math>2/x^2=54/x^3</math>.
+Solving, we obtain <math>x=27</math> as the critical value.  Hence, <math>k</math> has the maximum value of <math>(2*27-27)/27^2=1/27</math>.  Since <math>BP^2=405+729k</math>, the maximum value of <math>\overline {BP}</math> occurs at <math>k=1/27</math>, so <math>BP^2</math> has a maximum value of <math>405+729/27=\fbox{432}</math>.
+Note: Please edit this solution if it feels inadequate.
+~Shen Kislay Kai
-==Alternate Solution==
+===Solution 2===
-<asy>
+<center><asy>
-unitsize(4mm);
+unitsize(3mm);
+pair B=(0,13.5), C=(23.383,0);
+pair O=(7.794, 9), P=(2*7.794,0);
+pair T=(7.794,0), Q=(0,0);
+pair A=(2*7.794,4.5);
+draw(Q--B--C--Q);
+draw(O--T);
+draw(A--P);
+draw(Circle(O,9));
+dot(A);dot(B);dot(C);dot(T);dot(P);dot(O);dot(Q);
+label("\(B\)",B,NW);
+label("\(A\)",A,NE);
+label("\(O\)",O,N);
+label("\(P\)",P,S);
+label("\(T\)",T,S);
+label("\(Q\)",Q,S);
+label("\(C\)",C,E);
+label("\(\theta\)",C + (-1.7,-0.2), NW);
+label("\(9\)", (B+O)/2, N);
+label("\(9\)", (O+A)/2, N);
+label("\(9\)", (O+T)/2,W);
+</asy></center>
+From the diagram, we see that <math>BQ = OT + BO \sin\theta = 9 + 9\sin\theta = 9(1 + \sin\theta)</math>, and that <math>QP = BA\cos\theta = 18\cos\theta</math>.
+<cmath>\begin{align*}BP^2 &= BQ^2 + QP^2 = 9^2(1 + \sin\theta)^2 + 18^2\cos^2\theta\\
+&= 9^2[1 + 2\sin\theta + \sin^2\theta + 4(1 - \sin^2\theta)]\\
+BP^2 &= 9^2[5 + 2\sin\theta - 3\sin^2\theta]\end{align*}</cmath>
+This is a [[quadratic equation]], maximized when <math>\sin\theta = \frac { - 2}{ - 6} = \frac {1}{3}</math>. Thus, <math>m^2 = 9^2[5 + \frac {2}{3} - \frac {1}{3}] = \boxed{432}</math>.
+===Solution 3 (Calculus Bash)===
+<center><asy>
+unitsize(3mm);
 pair B=(0,13.5), C=(23.383,0);
 pair O=(7.794, 9), P=(2*7.794,0);
@@ Line 49: / Line 101: @@
 label("\(Q\)",Q,S);
 label("\(C\)",C,E);
-label("\(\theta\)",C + (-1.5,0), NW);
 label("\(9\)", (B+O)/2, N);
 label("\(9\)", (O+A)/2, N);
 label("\(9\)", (O+T)/2,W);
-</asy>
+</asy></center>
+(Diagram credit goes to Solution 2)
+We let <math>AC=x</math>. From similar triangles, we have that <math>PC=\frac{x\sqrt{x^2+18x}}{x+9}</math> (Use Pythagorean on <math>\triangle\omega TC</math> and then using <math>\triangle\omega CT\sim\triangle ACP</math>). Similarly, <math>TP=QT=\frac{9\sqrt{x^2+18x}}{x+9}</math>. Using the Pythagorean Theorem again and <math>\triangle CAP\sim\triangle CBQ</math>, <math>BQ=\sqrt{(x+18)^2-(\frac{(x+18)\sqrt{x^2+18x}}{x+9})^2}</math>. Using the Pythagorean Theorem <math>\bold{again}</math>, <math>BP=\sqrt{(x+18)^2-(\frac{(x+18)\sqrt{x^2+18x}}{x+9})^2+(\frac{18\sqrt{x^2+18x}}{x+9})^2}</math>. After a large bashful simplification, <math>BP=\sqrt{405+\frac{1458x-6561}{x^2+18x+81}}</math>. The fraction is equivalent to <math>729\frac{2x-9}{(x+9)^2}</math>. Taking the derivative of the fraction and solving for x, we get that <math>x=18</math>. Plugging <math>x=18</math> back into the expression for <math>BP</math> yields <math>\sqrt{432}</math>, so the answer is <math>(\sqrt{432})^2=\boxed{432}</math>.
+===Solution 4===
+<center><asy>
+unitsize(3mm);
+pair B=(0,13.5), C=(23.383,0);
+pair O=(7.794, 9), P=(2*7.794,0);
+pair T=(7.794,0), Q=(0,0);
+pair A=(2*7.794,4.5);
+draw(Q--B--C--Q);
+draw(O--T);
+draw(A--P);
+draw(Circle(O,9));
+dot(A);dot(B);dot(C);dot(T);dot(P);dot(O);dot(Q);
+label("\(B\)",B,NW);
+label("\(A\)",A,NE);
+label("\(\omega\)",O,N);
+label("\(P\)",P,S);
+label("\(T\)",T,S);
+label("\(Q\)",Q,S);
+label("\(C\)",C,E);
+label("\(9\)", (B+O)/2, N);
+label("\(9\)", (O+A)/2, N);
+label("\(9\)", (O+T)/2,W);
+</asy></center>
-<math>BQ = \omega T + B\omega\sin\theta = 9 + 9\sin\theta = 9(1 + \sin\theta)</math>
+(Diagram credit goes to Solution 2)
-<math>QP = BA\cos\theta = 18\cos\theta</math>
+Let <math>AC=x</math>. The only constraint on <math>x</math> is that it must be greater than <math>0</math>. Using similar triangles, we can deduce that <math>PA=\frac{9x}{x+9}</math>. Now, apply law of cosines on <math>\triangle PAB</math>. <cmath>BP^2=\left(\frac{9x^2}{x+9}\right)^2+18^2-2(18)\left(\frac{9x}{x+9}\right)\cos(\angle PAB).</cmath> We can see that <math>\cos(\angle PAB)=\cos(180^{\circ}-\angle PAC)=\cos(\angle PAC -90^{\circ})=-\sin(\angle PCA)</math>. We can find <math>-\sin(\angle PCA)=-\frac{9}{x+9}</math>. Plugging this into our equation, we get:
+<cmath>BP^2=\left(\frac{9x^2}{x+9}\right)^2+18^2-2(18)\left(\frac{9x}{x+9}\right)\left(-\frac{9}{x+9}\right).</cmath> Eventually, <cmath>BP^2 = 81\left(\frac{x^2+36x}{(x+9)^2}+4\right).</cmath> We want to maximize <math>\frac{x^2+36x}{(x+9)^2}</math>. There are many ways to maximize this expression, discussed here: https://artofproblemsolving.com/community/c4h2292700_maximization. The maximum result of that expression is <math>\frac{4}{3}</math>. Finally, evaluating <math>BP^2</math> for this value <math>81\left(\frac{4}{3}+4\right) = \boxed{432}</math>.
-<math>BP^2 = BQ^2 + QP^2 = 9^2(1 + \sin\theta)^2 + 18^2\cos^2\theta</math>
-<math>BP^2 = 9^2[1 + 2\sin\theta + \sin^2\theta + 4(1 - \sin^2\theta)]</math>
+~superagh
-<math>BP^2 = 9^2[5 + 2\sin\theta - 3\sin^2\theta]</math>
+===Solution 5 (Clean)===
+Let <math>h</math> be the distance from <math>A</math> to <math>CT</math>. Observe that <math>h</math> takes any value from <math>0</math> to <math>2r</math>, where <math>r</math> is the radius of the circle.
-Maximum at <math>\sin\theta = \frac { - 2}{ - 6} = \frac {1}{3}</math>
+Let <math>Q</math> be the foot of the altitude from <math>B</math> to <math>CT</math>. It is clear that <math>T</math> is the midpoint of <math>PQ</math>, and so the length <math>OT</math> is the average of <math>AP</math> and <math>BQ</math>. It follows thus that <math>BQ = 2r - h</math>.
-<math>m^2 = 9^2[5 + \frac {2}{3} - \frac {1}{3}] = \boxed{432}</math>
+We compute <math>PT = \sqrt{r^2 - (r - h)^2} = \sqrt{h(2r - h)},</math>
+and so <math>BP^2 = PQ^2 + BQ^2 = 4PT^2 + BQ^2 = 4h(2r - h) + (2r-h)^2 = (2r-h)(2r + 3h)</math>.
+This is <math>\frac{1}{3}(6r - 3h)(2r + 3h) \le \frac{1}{3} \cdot \left( \frac{8r}{2} \right)^2</math>. Equality is attained, so thus we extract the answer of <math>\frac{16 \cdot 9^2}{3} = 27 \cdot 16 = \boxed{432}.</math>
 == See also ==
@@ Line 73: / Line 158: @@
 [[Category:Intermediate Geometry Problems]]
+{{MAA Notice}}

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