Key Concepts and Formulas

• Equation of a Straight Line (Point-Slope Form): A line passing through the point $(x_1, y_1)$ with a slope $m$ can be represented as $y - y_1 = m(x - x_1)$.
• Intercepts with Coordinate Axes: The x-intercept is the point where the line crosses the x-axis (where $y=0$), and the y-intercept is the point where the line crosses the y-axis (where $x=0$).
• AM-GM Inequality: For non-negative real numbers $a$ and $b$, $\frac{a+b}{2} \ge \sqrt{ab}$. Equality holds when $a=b$.

Step-by-Step Solution

Step 1: Formulate the Equation of the Line and Determine Intercepts

First, we need to find the equation of the line and then determine the coordinates of points $A$ and $B$ where it intersects the axes.

• Equation of the Line: The line passes through the point $(x_1, y_1) = (4, -9)$ and has a slope $m$. Using the point-slope form, the equation of the line is: $y - (-9) = m(x - 4)$ $y + 9 = m(x - 4)$ Explanation: This step translates the given information (a point and a slope) into an algebraic equation representing the line.

• Coordinates of Point A (x-intercept): Point $A$ lies on the x-axis, so its y-coordinate is $0$. Substitute $y=0$ into the line's equation: $0 + 9 = m(x_A - 4)$ $9 = m(x_A - 4)$ Since $m > 0$, we can divide by $m$: $x_A - 4 = \frac{9}{m}$ $x_A = 4 + \frac{9}{m}$ So, point $A$ is $\left(4 + \frac{9}{m}, 0\right)$. Explanation: By setting $y=0$, we find the x-coordinate where the line crosses the x-axis. Since $m>0$, $4 + \frac{9}{m}$ will always be positive, meaning $A$ is on the positive x-axis.

• Coordinates of Point B (y-intercept): Point $B$ lies on the y-axis, so its x-coordinate is $0$. Substitute $x=0$ into the line's equation: $y_B + 9 = m(0 - 4)$ $y_B + 9 = -4m$ $y_B = -9 - 4m$ So, point $B$ is $(0, -9 - 4m)$. Explanation: By setting $x=0$, we find the y-coordinate where the line crosses the y-axis. Since $m>0$, $-4m$ is negative, making $-9 - 4m$ always negative. This means $B$ is on the negative y-axis.

Step 2: Express the Sum of Distances from the Origin in Terms of $m$

We need to find the sum of the distances of $A$ and $B$ from the origin $O(0,0)$. Let this sum be $S(m)$.

• Distance OA: $OA = \left|4 + \frac{9}{m}\right|$ Since $m > 0$, $4 + \frac{9}{m}$ is always positive. $OA = 4 + \frac{9}{m}$

• Distance OB: $OB = |-9 - 4m|$ Since $m > 0$, $-9 - 4m$ is always negative. Therefore, its absolute value is its negation: $OB = -(-9 - 4m) = 9 + 4m$

• Sum of Distances S(m): $S(m) = OA + OB = \left(4 + \frac{9}{m}\right) + (9 + 4m)$ $S(m) = 13 + 4m + \frac{9}{m}$ Explanation: Distances are always non-negative. We use the absolute value to ensure this. The expression for $S(m)$ is now a function of the slope $m$, which we need to minimize. The domain for $m$ is $m>0$.

Step 3: Optimize the Function $S(m)$ Using AM-GM Inequality

To find the minimum value of $S(m)$, we will use the AM-GM inequality on the terms $4m$ and $\frac{9}{m}$ (since $m>0$, both terms are positive).

$\frac{4m + \frac{9}{m}}{2} \ge \sqrt{4m \cdot \frac{9}{m}}$ $4m + \frac{9}{m} \ge 2\sqrt{36}$ $4m + \frac{9}{m} \ge 2 \times 6$ $4m + \frac{9}{m} \ge 12$

The equality holds when $4m = \frac{9}{m}$, which gives $4m^2 = 9 \Rightarrow m = \frac{3}{2}$.

Thus, the minimum value of $4m + \frac{9}{m}$ is $12$.

Explanation: AM-GM provides an elegant way to find the minimum of sums of positive terms, especially when their product is constant. It confirms the critical value of $m$ and the minimum value of the variable part of $S(m)$.

Step 4: Calculate the Minimum Value of $S(m)$

Substitute the optimal value of $4m + \frac{9}{m} = 12$ back into the expression for $S(m)$.

$S(m) = 13 + 4m + \frac{9}{m} = 13 + 12 = 25$

Therefore, the minimum value of the sum of the distances of $A$ and $B$ from the origin is $25$.

Common Mistakes & Tips

• Absolute Values for Distances: Always remember that distances are non-negative.
• Domain of $m$: The condition $m > 0$ is crucial.
• AM-GM Applicability: Recognize when AM-GM can be used for quick optimization.

Summary

This problem demonstrates a classic application of coordinate geometry and the AM-GM inequality. The key steps involve: setting up the equation of the line, finding the x and y intercepts in terms of the slope $m$, formulating the total distance function $S(m)$, and minimizing $S(m)$ using the AM-GM inequality.

The minimum sum of distances is $\boxed{25}$, which corresponds to option (D). The final answer is \boxed{25}.