Difference between revisions of "Mock AIME 3 2006-2007 Problems/Problem 11"

(Problem)
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If <math>x</math> and <math>y</math> are real numbers such that <math>2xy+2x^2=6+x^2+y^2</math> find the minimum value of <math>(x^2+y^2)^2</math>.
 
If <math>x</math> and <math>y</math> are real numbers such that <math>2xy+2x^2=6+x^2+y^2</math> find the minimum value of <math>(x^2+y^2)^2</math>.
  
== Solution ==
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== Solution 1 ==
  
  
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So the desired minimum is <math>\frac{36}{2}=\boxed{018}</math>
 
So the desired minimum is <math>\frac{36}{2}=\boxed{018}</math>
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== Solution 2 ==
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Since <math>x^2 + y^2 \ge 0</math>,finding the minimum value of <math>(x^2 + y^2)^2</math> is similar to finding that of <math>x^2 + y^2</math>. Let <math>x^2 + y^2 = a</math>, where <math>a</math> is the minimum value. We can rewrite this as <math>y^2 = -x^2 + a</math> and <math>y = \sqrt{-x^2 + a}</math>.
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<cmath>2xy + 2x^2 = x^2 + y^2 + 6</cmath><cmath>2x(\sqrt{-x^2 + a}) + 2x^2 = x^2 + (-x^2 + a) + 6</cmath><cmath>2x(\sqrt{-x^2 + a}) + 2x^2 = a + 6</cmath><cmath>2x^2 - (a + 6) = -2x(\sqrt{x^2 + a})</cmath>.
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<cmath>4x^2 - 4(a + 6)x^2 + (a + 6)^2 = 4x^2(-x^2 + a)</cmath>.
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<cmath>8x^4 - 8(a + 3)x^2 + (a + 6)^2 = 0</cmath>.
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We want this polynomial to factor in the form <math>(x^2 - r)(x^2 - s)</math>, where at least one of <math>r, s \ge 0</math>. ( If <math>r, s < 0</math>, the equations <math>x^2 = r</math> and <math>x^2 = s</math> would have no real solutions). Since <math>a > 0</math>, both <math>-8(a + 3) > 0</math> and <math>(a + 6)^2 > 0</math>, so <math>r, s > 0</math>.
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We can now use the “discriminant” to determine acceptable values of <math>a</math>. <math>(8(a + 3))^2 - 4\cdot 8 \cdot (a + 6)^2 \ge 0</math> simplifies to <math>a^2 \ge 18</math>.
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Since <math>a^2 = (x^2 + y^2)^2</math>, the minimum value of <math>(x^2 + y^2)^2 = \boxed{18}</math>.

Revision as of 14:15, 8 November 2019

Problem

If $x$ and $y$ are real numbers such that $2xy+2x^2=6+x^2+y^2$ find the minimum value of $(x^2+y^2)^2$.

Solution 1

Factoring the LHS gives $2x(x+y)=6+x^2+y^2$.

Now converting to polar: $2r\cos(a)(r\cos(a)+r\sin(a))=6+r^2$

$2\cos(a)(\cos(a)+\sin(a))=\frac{6}{r^2}+1$

$2\cos^2(a)-1+2\cos(a)\sin(a)=\frac{6}{r^2}$

$\cos(2a)+\sin(2a)=\frac{6}{r^2}$

$r^2=\frac{6}{\cos(2a)+\sin(2a)}$

Since we want to find $(x^2+y^2)^2=r^4$, $r^4=\frac{36}{(\cos(2a)+\sin(2a))^2}=\frac{36}{1+2\cos(2a)\sin(2a)}=\frac{36}{1+\sin(4a)}$

Since we want the minimum of this expression, we need to maximize the denominator. The maximum of the sine function is 1

(one value of $a$ which produces this maximum is $a=\frac{\pi}{4}$)

So the desired minimum is $\frac{36}{2}=\boxed{018}$

Solution 2

Since $x^2 + y^2 \ge 0$,finding the minimum value of $(x^2 + y^2)^2$ is similar to finding that of $x^2 + y^2$. Let $x^2 + y^2 = a$, where $a$ is the minimum value. We can rewrite this as $y^2 = -x^2 + a$ and $y = \sqrt{-x^2 + a}$. \[2xy + 2x^2 = x^2 + y^2 + 6\]\[2x(\sqrt{-x^2 + a}) + 2x^2 = x^2 + (-x^2 + a) + 6\]\[2x(\sqrt{-x^2 + a}) + 2x^2 = a + 6\]\[2x^2 - (a + 6) = -2x(\sqrt{x^2 + a})\]. \[4x^2 - 4(a + 6)x^2 + (a + 6)^2 = 4x^2(-x^2 + a)\]. \[8x^4 - 8(a + 3)x^2 + (a + 6)^2 = 0\]. We want this polynomial to factor in the form $(x^2 - r)(x^2 - s)$, where at least one of $r, s \ge 0$. ( If $r, s < 0$, the equations $x^2 = r$ and $x^2 = s$ would have no real solutions). Since $a > 0$, both $-8(a + 3) > 0$ and $(a + 6)^2 > 0$, so $r, s > 0$.

We can now use the “discriminant” to determine acceptable values of $a$. $(8(a + 3))^2 - 4\cdot 8 \cdot (a + 6)^2 \ge 0$ simplifies to $a^2 \ge 18$. Since $a^2 = (x^2 + y^2)^2$, the minimum value of $(x^2 + y^2)^2 = \boxed{18}$.

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