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

Revision as of 14:48, 15 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 QM-GM Inequality)

$(x-3)^2 + (y-4)^2 = h + 25$ $(x-5)^2 + (y+2)^2 = k + 29$ Distance between 2 circle centers is $(O_{1}O_{2})^2 = (5-3)^2 + (4 - (-2)) ^2 = 40$ the 2 circle must intersect given there exists one or more pair of (x,y), connecting $O_{1}O_{2}$ and any one of the 2 circle intersection point we get a triangle with 3 sides ( radius ( $O_{1}$ ) , radius ( $O_{2}$ ) , $O_{1}O_{2}$ ) , then $radius (O_{1}) + radius (O_{2}) \geq O_{1}O_{2}$ $\sqrt{h+25} + \sqrt{k+29} \geq 2*\sqrt{10}$ the equal sign will be reached when 2 circles are external tangent to each other,

Apply QM-GM inequality $2(a^2 + b^2) \geq (a+b)^2$ in step 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

~Kathan

@@ Line 29: / Line 29: @@
 ~mitsuihisashi14
-==Solution 2 (Coordinate Geometry and HM-GM)==
+==Solution 2 (Coordinate Geometry and QM-GM Inequality)==
 [[Image:2024_amc_12B_P13.PNG|thumb|center|400px|]]
@@ Line 35: / Line 35: @@
 <cmath>(x-3)^2 + (y-4)^2 = h + 25 </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>(O_{1}O_{2})^2 = (5-3)^2 + (4 - (-2)) ^2 = 40 </cmath>
-the 2 circle must interact, so
+the 2 circle must intersect given there exists one or more pair of (x,y), connecting <math>O_{1}O_{2}</math> and any one of the 2 circle intersection point we get a triangle with 3 sides ( radius (<math>O_{1}</math>) , radius (<math>O_{2}</math>) , <math>O_{1}O_{2}</math>) , then
+<cmath> radius (O_{1}) + radius (O_{2}) \geq O_{1}O_{2} </cmath>
 <cmath>\sqrt{h+25} + \sqrt{k+29}   \geq  2*\sqrt{10} </cmath>
-the min value will be reached when 2 circles are external tangent to each other
+the equal sign will be reached when 2 circles are external tangent to each other,
+Apply QM-GM inequality <math> 2(a^2 + b^2) \geq (a+b)^2</math> in step below, we get
 <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}
 \geq   \frac{\left(2\sqrt{10}\right)^2}{2} = 20.
 </cmath>
-the above step uses HM-GM inequality <math> 2(a^2 + b^2) \geq (a+b)^2 </math>
-min( h + k ) = <math>\boxed{C -34} </math>.
+Therefore,  h + k <math> \geq 20 - 54 = \boxed{C -34} </math>.

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