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Difference between revisions of "2024 AMC 12B Problems/Problem 13"

(→‎Solution 2 (Coordinate Geometry and HM-GM))
(→‎Solution 2 (Coordinate Geometry and AM-QM Inequality))
 
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==Solution 1 (Easy and Fast)==
 
==Solution 1 (Easy and Fast)==
  
−
Adding up the first and second statement, we get h+k with:
+
Adding up the first and second equation, we get:
−
 
+
<cmath>
−
= 2x^2 + 2y^2 - 16x - 4y
+
\begin{align*}
−
 
+
h + k &= 2x^2 + 2y^2 - 16x - 4y \\
−
= 2(x^2 - 8x) + 2(y^2 - 2y)
+
&= 2(x^2 - 8x) + 2(y^2 - 2y) \\
 +
&= 2(x^2 - 8x) + 2(y^2 - 2y) \\
 +
&= 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1) \\
 +
&= 2(x - 4)^2 + 2(y - 1)^2 - 34
 +
\end{align*}
 +
</cmath>
 +
All squared values must be greater than or equal to <math>0</math>. As we are aiming for the minimum value, we set the two squared terms to be <math>0</math>.
  
−
= 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1)
+
This leads to <math>\min(h + k) = 0 + 0 - 34 = \boxed{\textbf{(C)} -34}</math>
−
 
 
−
= 2(x - 4)^2 + 2(y - 1)^2 - 34
 
−
 
 
−
All squared values must be greater or equal to 0. As we are aiming for the minimum value, we let the 2 squared terms be 0.
 
−
 
 
−
This leads to (h+k)min = 0 + 0 - 34 = (C) -34
 
  
 
~mitsuihisashi14
 
~mitsuihisashi14
  
−
==Solution 2 (Coordinate Geometry and HM-GM)==
+
==Solution 2 (Coordinate Geometry and AM-QM Inequality)==
  
−
[[Image:2024_amc_12B_P13.PNG|thumb|center|400px|]]
+
[[Image:2024_amc_12B_P13_V2.PNG|thumb|center|500px|]]
  
 
<cmath>(x-3)^2 + (y-4)^2 = h + 25 </cmath>
 
<cmath>(x-3)^2 + (y-4)^2 = h + 25 </cmath>
 
<cmath>(x-5)^2 + (y+2)^2 = k + 29 </cmath>
 
<cmath>(x-5)^2 + (y+2)^2 = k + 29 </cmath>
−
distance between 2 circle centers is <cmath>d^2 = (5-3)^2 + (4 - (-2)) ^2 = 40 </cmath>
+
The distance between 2 circle centers is <cmath>(O_{1}O_{2})^2 = (5-3)^2 + (4 - (-2)) ^2 = 40 </cmath>
−
<cmath>\sqrt{h+25} + \sqrt{k+29}  = 2*\sqrt{10} </cmath>
+
The 2 circles must intersect given there exists one or more pairs of (x,y), connecting <math>O_{1}O_{2}</math> and any pair of the 2 circle intersection points gives us a triangle with 3 sides, then   
 +
<cmath> radius (O_{1}) + radius (O_{2}) \geq O_{1}O_{2} </cmath>  
 +
<cmath>\sqrt{h+25} + \sqrt{k+29}  \geq  2\cdot\sqrt{10} </cmath>
 +
Note that they will be equal if and only if the circles are tangent,
 +
 
 +
Applying the AM-QM inequality (<math> 2(a^2 + b^2) \geq (a+b)^2</math>) in the steps below, we get 
 
<cmath>
 
<cmath>
 
   h + k + 54 = (h + 25) + (k + 29) =\sqrt{(h + 25)}^2 + \sqrt{(k + 29)}^2 \geq \frac{\left(\sqrt{h + 25} + \sqrt{k + 29}\right)^2}{2}
 
   h + k + 54 = (h + 25) + (k + 29) =\sqrt{(h + 25)}^2 + \sqrt{(k + 29)}^2 \geq \frac{\left(\sqrt{h + 25} + \sqrt{k + 29}\right)^2}{2}
−
=  \frac{\left(2\sqrt{10}\right)^2}{2} = 20.
+
\geq  \frac{\left(2\sqrt{10}\right)^2}{2} = 20.
 
</cmath>
 
</cmath>
−
min( h + k ) = <math>\boxed{C -34} </math>.   
+
 
 +
Therefore, <math>h + k \geq 20 - 54 = \boxed{C -34} </math>.   
  
  
 
~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
 
~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
 +
 +
==Solution 3==
 +
[[Image: 2024_AMC_12B_P13.jpeg|thumb|center|600px|]]
 +
~Kathan
 +
 +
==Video Solution 1 by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=U0PqhU73yU0
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2024|ab=B|num-b=12|num-a=14}}
 
{{AMC12 box|year=2024|ab=B|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 20:30, 17 November 2024

Problem 13

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations\[x^2 + y^2 - 6x - 8y = h\]\[x^2 + y^2 - 10x + 4y = k\]What is the minimum possible value of $h+k$?

$\textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad$


Solution 1 (Easy and Fast)

Adding up the first and second equation, we get: \begin{align*} h + k &= 2x^2 + 2y^2 - 16x - 4y \\ &= 2(x^2 - 8x) + 2(y^2 - 2y) \\ &= 2(x^2 - 8x) + 2(y^2 - 2y) \\ &= 2(x^2 - 8x + 16) - (2)(16) + 2(y^2 - 2y + 1) - (2)(1) \\ &= 2(x - 4)^2 + 2(y - 1)^2 - 34 \end{align*} All squared values must be greater than or equal to $0$. As we are aiming for the minimum value, we set the two squared terms to be $0$.

This leads to $\min(h + k) = 0 + 0 - 34 = \boxed{\textbf{(C)} -34}$

~mitsuihisashi14

Solution 2 (Coordinate Geometry and AM-QM Inequality)

2024 amc 12B P13 V2.PNG

\[(x-3)^2 + (y-4)^2 = h + 25\] \[(x-5)^2 + (y+2)^2 = k + 29\] The distance between 2 circle centers is \[(O_{1}O_{2})^2 = (5-3)^2 + (4 - (-2)) ^2 = 40\] The 2 circles must intersect given there exists one or more pairs of (x,y), connecting $O_{1}O_{2}$ and any pair of the 2 circle intersection points gives us a triangle with 3 sides, then \[radius (O_{1}) + radius (O_{2}) \geq O_{1}O_{2}\] \[\sqrt{h+25} + \sqrt{k+29} \geq 2\cdot\sqrt{10}\] Note that they will be equal if and only if the circles are tangent,

Applying the AM-QM inequality ($2(a^2 + b^2) \geq (a+b)^2$) in the steps below, we get \[h + k + 54 = (h + 25) + (k + 29) =\sqrt{(h + 25)}^2 + \sqrt{(k + 29)}^2 \geq \frac{\left(\sqrt{h + 25} + \sqrt{k + 29}\right)^2}{2} \geq \frac{\left(2\sqrt{10}\right)^2}{2} = 20.\]

Therefore, $h + k \geq 20 - 54 = \boxed{C -34}$.


~luckuso

Solution 3

2024 AMC 12B P13.jpeg

~Kathan

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=U0PqhU73yU0

See also

2024 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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All AMC 12 Problems and Solutions

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