Key Concepts and Formulas

• Intercept Form of a Straight Line: A line with x-intercept 'a' and y-intercept 'b' can be written as $\frac{x}{a} + \frac{y}{b} = 1$.
• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.

Step-by-Step Solution

Step 1: Determine the Intercepts of the Given Line

We are given the equation of the family of lines as: $x \cos \theta+y \sin \theta=7$ To find the x and y intercepts, we rewrite the equation in intercept form. Divide both sides of the equation by 7: $\frac{x \cos \theta}{7}+\frac{y \sin \theta}{7}=1$ Rearrange to get the intercept form: $\frac{x}{\left(\frac{7}{\cos \theta}\right)}+\frac{y}{\left(\frac{7}{\sin \theta}\right)}=1$ The x-intercept is $\frac{7}{\cos \theta}$ and the y-intercept is $\frac{7}{\sin \theta}$.

Step 2: Identify the Endpoints of the Line Segment

The line segment is bounded by the coordinate axes. The x-intercept is the point where the line intersects the x-axis, and the y-intercept is the point where the line intersects the y-axis. Therefore, the endpoints of the line segment are: $A = \left(\frac{7}{\cos \theta}, 0\right)$ $B = \left(0, \frac{7}{\sin \theta}\right)$

Step 3: Find the Coordinates of the Midpoint

Let $M(h, k)$ be the midpoint of the line segment $AB$. We use the midpoint formula to find the coordinates of $M$: $h = \frac{\frac{7}{\cos \theta} + 0}{2} = \frac{7}{2 \cos \theta}$ $k = \frac{0 + \frac{7}{\sin \theta}}{2} = \frac{7}{2 \sin \theta}$

Step 4: Use the Given Point to Determine the Value of $\theta$

We are given that the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by these midpoints. Therefore, we can equate the coordinates of the given point with the expressions for $h$ and $k$: $h = \alpha$ $k = \frac{7 \sqrt{3}}{3}$ Substitute the value of $k$ into the midpoint equation for $k$: $\frac{7 \sqrt{3}}{3} = \frac{7}{2 \sin \theta}$ Solve for $\sin \theta$: $\sin \theta = \frac{7}{2} \cdot \frac{3}{7 \sqrt{3}} = \frac{3}{2 \sqrt{3}} = \frac{\sqrt{3}}{2}$ Since $\theta \in\left(0, \frac{\pi}{2}\right)$, the only angle for which $\sin \theta = \frac{\sqrt{3}}{2}$ is $\theta = \frac{\pi}{3}$.

Step 5: Calculate the Value of $\alpha$

Now that we have the value of $\theta = \frac{\pi}{3}$, we can substitute it into the expression for $h$ (which is equal to $\alpha$): $\alpha = \frac{7}{2 \cos \theta}$ Substitute $\theta = \frac{\pi}{3}$: $\alpha = \frac{7}{2 \cos \left(\frac{\pi}{3}\right)}$ Since $\cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$: $\alpha = \frac{7}{2 \cdot \frac{1}{2}} = \frac{7}{1} = 7$

Common Mistakes & Tips

• Midpoint Formula: Double-check that you are correctly applying the midpoint formula, especially the division by 2.
• Domain of $\theta$: The restriction on $\theta$ is essential for finding a unique solution for $\theta$ and ensuring intercepts are positive.
• Algebraic Manipulation: Pay close attention to simplifying expressions, especially those involving fractions and radicals.

Summary

The problem required finding the x-coordinate of a point on the locus of midpoints of line segments formed by the intersection of a family of lines with the coordinate axes. We first found the intercepts of the line, then calculated the midpoint coordinates in terms of $\theta$. By equating the given y-coordinate with the midpoint's y-coordinate, we solved for $\theta$. Finally, substituting this value of $\theta$ into the expression for the midpoint's x-coordinate yielded the value of $\alpha$.

The final answer is $\boxed{7}$, which corresponds to option (B).