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√3
• C. 7
• D. 7√3

Hint:

To find the locus of midpoints, first find the general coordinates of the midpoint in terms of a parameter (like \( \theta \) in this case). Then, eliminate the parameter to get the equation of the locus.

Answer 1

The Correct Option is C

The given equation of the line is:

\( x\cos\theta + y\sin\theta = 7 \)

Step 1: Intercepts of the line

The x-intercept is: \( \frac{7}{\cos\theta} \)

The y-intercept is: \( \frac{7}{\sin\theta} \)

Thus, the intercept points are:

\( A \left( \frac{7}{\cos\theta} , 0 \right) \) and \( B \left( 0, \frac{7}{\sin\theta} \right) \)

Step 2: Coordinates of the midpoint \(M(h,k)\)

The midpoint of \( A \) and \( B \) is given by:

\( h = \frac{7}{2\cos\theta}, \quad k = \frac{7}{2\sin\theta} \)

Step 3: Solving for \( \theta \)

From the given information:

\( k = \frac{7\sqrt{3}}{3} \)

Substitute \( k = \frac{7}{2\sin\theta} \):

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

Thus:

\( \theta = \frac{\pi}{3} \)

Step 4: Solving for \( \alpha \)

Using \( h = \frac{7}{2\cos\theta} \):

\( \alpha = \frac{7}{2\cos\theta} = \frac{7}{2\cos(\frac{\pi}{3})} = \frac{7}{2(\frac{1}{2})} = 7 \)