1996 AHSME Problems/Problem 25

Problem

Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?

$\text{(A)}\ 72\qquad\text{(B)}\ 73\qquad\text{(C)}\ 74\qquad\text{(D)}\ 75\qquad\text{(E)}\ 76$

Solution 1

Complete the square to get $$(x-7)^2 + (y-3)^2 = 64.$$ Applying Cauchy-Schwarz directly, $$64*25=(3^2+4^2)((x-7)^2 + (y-3)^2) \ge (3(x-7)+4(y-3))^2.$$ $$40 \ge 3x+4y-33$$ $$3x+4y \le 73.$$ Thus our answer is $\boxed{(B)}$.

Solution 2 (Geometric)

The first equation is a circle, so we find its center and radius by completing the square: $x^2 - 14x + y^2 - 6y = 6$, so $$(x-7)^2 + (y-3)^2 = (x^2- 14x + 49) + (y^2 - 6y + 9) = 6 + 49 + 9 = 64.$$

So we have a circle centered at $(7,3)$ with radius $8$, and we want to find the max of $3x + 4y$.

The set of lines $3x + 4y = A$ are all parallel, with slope $-\frac{3}{4}$. Increasing $A$ shifts the lines up and/or to the right.

We want to shift this line up high enough that it's tangent to the circle, but not so high that it misses the circle altogether. This means $3x + 4y = A$ will be tangent to the circle.

Imagine that this line hits the circle at point $(a,b)$. The slope of the radius connecting the center of the circle, $(7,3)$, to tangent point $(a,b)$ will be $\frac{4}{3}$, since the radius is perpendicular to the tangent line.

So we have a point, $(7,3)$, and a slope of $\frac{4}{3}$ that represents the slope of the radius to the tangent point. Let's start at the point $(7,3)$. If we go $4k$ units up and $3k$ units right from $(7,3)$, we would arrive at a point that's $5k$ units away. But in reality we want $5k = 8$ to reach the tangent point, since the radius of the circle is $8$.

Thus, $k = \frac{8}{5}$, and we want to travel $4\cdot \frac{8}{5}$ up and $3\cdot \frac{8}{5}$ over from the point $(7,3)$ to reach our maximum. This means the maximum value of $3x + 4y$ occurs at $\left(7 +3\cdot \frac{8}{5}, 3 + 4\cdot \frac{8}{5}\right)$, which is $\left(\frac{59}{5}, \frac{47}{5}\right).$

Plug in those values for $x$ and $y$, and you get the maximum value of $3x + 4y = 3\cdot\frac{59}{5} + 4\cdot\frac{47}{5} = \boxed{73}$, which is option $\boxed{(B)}$.

Solution 2B

Let the tangent point be $P$, and the tangent line's x-intercept be $Q$. Consider the horizontal line starting from center of circle (O) meeting the tangent line at K. Now triangle $OPK$ is 3-4-5, $OP=8$, so $OK = \frac{5}{3}*8 = \frac{40}{3}$. Note that the horizontal distance from $O$ to the origin is $7$, and the horizontal distance from K to Q is 4, ($\frac{4}{3}$ of its y coordinate), so the x-intercept is $7+4+OK = 73/3$. The value of $3x+4y$ is 73 at point $Q$. Note that this value is constant on the tangent line, so there is no need to calculate the coordinate of $P$. $\boxed{(B)}$.

Solution 3

Let $z = 3x + 4y$. Solving for $y$, we get $y = (z - 3x)/4$. Substituting into the given equation, we get $$x^2 + \left( \frac{z - 3x}{4} \right)^2 = 14x + 6 \cdot \frac{z - 3x}{4} + 6,$$ which simplifies to $$25x^2 - (6z + 152)x + (z^2 - 24z - 96) = 0.$$

This quadratic equation has real roots in $x$ if and only if its discriminant is nonnegative, so $$(6z + 152)^2 - 4 \cdot 25 \cdot (z^2 - 24z - 96) \ge 0,$$ which simplifies to $$-64z^2 + 4224z + 32704 \ge 0,$$ which can be factored as $$-64(z + 7)(z - 73) \ge 0.$$ The largest value of $z$ that satisfies this inequality is $\boxed{73}$, which is $\boxed{(B)}$.

Solution 4 (Using Answer Choice + Calculus)

Implicitly differentiating the given equation with respect to $x$ yields:

$2x + 2y\frac{dy}{dx} = 14 + 6\frac{dy}{dx}$

Now solve for $\frac{dy}{dx}$ to obtain:

$\frac{dy}{dx} = -\frac{x - 7}{y - 3}$

Set the equation equal to zero to find the maximum occurs at $x = 7$

Plug this back into the equation that we are trying to maximize and see that we are left with: $21 + 4y$.

The only answer choice that can be obtained from this equation is $\bf{73}$

$aopsswag$