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

m (→‎Solution 1 (Easy and Fast))
(→‎Solution 2 (Coordinate Geometry and HM-GM))
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<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>
 
distance between 2 circle centers is <cmath>d^2 = (5-3)^2 + (4 - (-2)) ^2 = 40 </cmath>
−
<cmath>\sqrt{h+25} + \sqrt{k+29}  = 2*\sqrt{10} </cmath>
+
the 2 circle must interact, so
 +
<cmath>\sqrt{h+25} + \sqrt{k+29}  >= 2*\sqrt{10} </cmath>
 +
the min value will be reached when 2 circles are external tangent to each other
 
<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.
+
>=  \frac{\left(2\sqrt{10}\right)^2}{2} = 20.
 
</cmath>
 
</cmath>
 
min( h + k ) = <math>\boxed{C -34} </math>.   
 
min( h + k ) = <math>\boxed{C -34} </math>.   

Revision as of 20:43, 14 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 HM-GM)

2024 amc 12B P13.PNG

\[(x-3)^2 + (y-4)^2 = h + 25\] \[(x-5)^2 + (y+2)^2 = k + 29\] distance between 2 circle centers is \[d^2 = (5-3)^2 + (4 - (-2)) ^2 = 40\] the 2 circle must interact, so \[\sqrt{h+25} + \sqrt{k+29} >= 2*\sqrt{10}\] the min value will be reached when 2 circles are external tangent to each other \[h + k + 54 = (h + 25) + (k + 29) =\sqrt{(h + 25)}^2 + \sqrt{(k + 29)}^2 \geq \frac{\left(\sqrt{h + 25} + \sqrt{k + 29}\right)^2}{2} >= \frac{\left(2\sqrt{10}\right)^2}{2} = 20.\] min( h + k ) = $\boxed{C -34}$.


~luckuso

Solution 3

2024 AMC 12B P13.jpeg

~Kathan

See also

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