# 1984 AHSME Problems/Problem 29

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## Problem

Find the largest value for $\frac{y}{x}$ for pairs of real numbers $(x, y)$ which satisfy $(x-3)^2+(y-3)^2=6$.

$\mathrm{(A) \ }3+2\sqrt{2} \qquad \mathrm{(B) \ }2+\sqrt{3} \qquad \mathrm{(C) \ } 3\sqrt{3} \qquad \mathrm{(D) \ }6 \qquad \mathrm{(E) \ } 6+2\sqrt{3}$

## Solution

Let $\frac{y}{x}=k$, so that $y=kx$. Substituting this into the given equation $(x-3)^2+(y-3)^2=6$ yields $(x-3)^2+(kx-3)^2=6$. Multiplying this out and forming it into a quadratic yields $(1+k^2)x^2+(-6-6k)x+12=0$.

We want $x$ to be a real number, so we must have the discriminant $\geq0$. The discriminant is $(-6-6k)^2-4(1+k^2)(12)=36k^2+72k+36-48-48k^2=-12k^2+72k-12$. Therefore, we must have $-12k^2+72k-12\geq0$, or $k^2-6k+1\leq0$. The roots of this quadratic, using the quadratic formula, are $3\pm2\sqrt{2}$, so the quadratic can be factored as $(k-(3-2\sqrt{2}))(k-(3+2\sqrt{2}))\leq0$. We can now separate this into $3$ cases:

Case 1: $k<3-2\sqrt{2}$ Then, both terms in the factored quadratic are negative, so the inequality doesn't hold.

Case 2: $3-2\sqrt{2} Then, the first term is positive and the second is negative and the second is positive, so the inequality holds.

Case 3: $k>3+2\sqrt{2}$ Then, both terms are positive, so the inequality doesn't hold.

Also, when $k=3-2\sqrt{2}$ or $k=3+2\sqrt{2}$, the equality holds.

Therefore, we must have $3-2\sqrt{2}\leq k\leq3+2\sqrt{2}$, and the maximum value of $k=\frac{y}{x}$ is ${3+2\sqrt{2}}, \boxed{\text{A}}$.

## Solution 2

The equation represents a circle of radius $\sqrt{6}$ centered at $A=(3,3)$. To find the maximal $k$ with $y=kx$ is equivalent to finding the maximum slope of a line passing through the origin $O$ and intersecting the circle. The steepest such line is tangent to the circle at some point $X$. We have $XA=\sqrt{6}$, $OA=\sqrt{18}$, $\angle OXA = 90$ because the line is tangent to the circle. Using the pythagorean theorem, we have $OX=\sqrt{12}$.

The slope we are looking for is equivalent to $\tan (\theta + 45)$ where $\angle AOX = \theta$. Using tangent addition,

$$\tan (\theta + 45)= \frac{\tan \theta + \tan 45}{1-\tan\theta\tan 45} = \frac{\frac{1}{\sqrt{2}}+1}{1-\frac{1}{\sqrt{2}}}=3+2\sqrt{2}$$

So $\boxed{A}$ is the answer

## Solution 3

Following the steps of Solution 2, we get $OX=\sqrt{12}$. Let $X = (x, y)$. Based on the distance to the origin and to the centre of the circle

$$x^2 + y^2 = 12$$

$$(x - 3)^2 + (y - 3)^2 = 6$$

Expanding the latter and substituting the former into the equation gives $30 - 6(x + y) = 6$. $x + y = 4$. We can square this and remove the 1st equation to get $2xy = 4$, or $xy = 2$.

Substitute $y = 4 - x$ to get

$$x(4 - x) = 2$$

$$x^2 - 4x + 2 = 0$$

$$x = 2 \pm \sqrt{2}$$

Since $x$ and $y$ are symmetric, we want $x$ to be the lower number to maximize $k$, so $x = 2 - \sqrt{2}$ and $y = 2 + \sqrt{2}$.

$$\frac{y}{x} = \frac{2 + \sqrt{2}}{2 - \sqrt{2}}$$

$$\frac{y}{x} = \frac{(2 + \sqrt{2})^2}{2}$$

$$\frac{y}{x} = 3 + 2\sqrt{2}$$