Question

If the point \((α ,\frac{7√3}{3} )\) lies on the curve traced by the mid-points of the line segments of the lines x cosθ + y sinθ=7,θ∈( \(0,\frac{π}{2}\) ) between the coordinates axes, then α is equal to 

• A. -7
• B. \(-7\sqrt3\)
• C. 7
• D. \(7\sqrt3\)

Answer 1

The Correct Option is C

To find the value of \( \alpha \) such that the point \((\alpha, \frac{7\sqrt{3}}{3})\) lies on the curve traced by the mid-points of the line segments of the line \(x \cos \theta + y \sin \theta = 7\) between the coordinate axes, follow the steps below:

1. Consider the line \(x \cos \theta + y \sin \theta = 7\). This line intersects the x-axis and y-axis. To find the intersection points, set \(y = 0\) to find \(x\), and \(x = 0\) to find \(y\).
2. When \(y = 0\), the equation becomes:

   x \cos \theta = 7 \Rightarrow x = \frac{7}{\cos \theta}

   This gives the x-intercept.
3. When \(x = 0\), the equation becomes:

   y \sin \theta = 7 \Rightarrow y = \frac{7}{\sin \theta}

   This gives the y-intercept.
4. The mid-point of the line segment joining these intercepts is:

   \left( \frac{\frac{7}{\cos \theta} + 0}{2}, \frac{0 + \frac{7}{\sin \theta}}{2} \right) = \left( \frac{7}{2 \cos \theta}, \frac{7}{2 \sin \theta} \right)

5. We are given that this mid-point lies at \((\alpha, \frac{7\sqrt{3}}{3})\), meaning:

   \frac{7}{2 \sin \theta} = \frac{7\sqrt{3}}{3}

   Simplify to solve for \(\sin \theta\):

   \frac{7}{2 \sin \theta} = \frac{7\sqrt{3}}{3} \Rightarrow 2 \sin \theta = \frac{3}{\sqrt{3}} \Rightarrow \sin \theta = \frac{\sqrt{3}}{2}

6. At \(\sin \theta = \frac{\sqrt{3}}{2}\), \(\theta\) corresponds to an angle of \(60^\circ\) or \(\frac{\pi}{3}\). Hence, \(\cos \theta = \frac{1}{2}\).
7. Now, substitute \(\cos \theta = \frac{1}{2}\) into \(\alpha = \frac{7}{2 \cos \theta}\):

   \alpha = \frac{7}{2 \cdot \frac{1}{2}} = 7

Thus, the correct value of \(\alpha\) is 7.