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

Let's analyze the problem step by step and find out the value of \(\alpha\).

The line equation given is \(x \cos \theta + y \sin \theta = 7\), where \(\theta \in (0, \frac{\pi}{2})\). This represents a line in the first quadrant making angles with x-axis and y-axis depending on the value of \(\theta\).

The line segment discussed is between the x-axis and y-axis. For such a line segment, the x-intercept is obtained by setting \(y=0\):

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

Similarly, the y-intercept is obtained by setting \(x=0\):

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

The mid-point of the line segment joining these intercepts on the coordinate axes is found by averaging their coordinates. Thus, the mid-point \((x_m, y_m)\) is:

\(x_m = \frac{1}{2}\left(0 + \frac{7}{\cos \theta}\right) = \frac{7}{2\cos \theta}\)

\(y_m = \frac{1}{2}\left(\frac{7}{\sin \theta} + 0\right) = \frac{7}{2\sin \theta}\)

According to the problem, this point \((x_m, y_m)\) is given as \((\alpha, \frac{7\sqrt{3}}{3})\).

Equating the y-coordinate:

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

Solving for \(\sin \theta\):

\(\frac{1}{2\sin \theta} = \frac{\sqrt{3}}{3} \Rightarrow 2\sin \theta = \frac{3}{\sqrt{3}} = \sqrt{3} \Rightarrow \sin \theta = \frac{\sqrt{3}}{2}\)

The angle \(\theta\) which satisfies this is \(\theta = \frac{\pi}{3}\) as \(\sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}\).

Now, evaluating \(x_m\) when \(\cos \theta\) at this angle:

\(\cos \theta = \frac{1}{2}\) when \(\theta = \frac{\pi}{3}\).

Thus, \(x_m = \frac{7}{2 \times \frac{1}{2}} = 7\).

The value of \(\alpha\) from \((\alpha, \frac{7\sqrt{3}}{3})\) becomes 7, which matches one of the given options.

Thus, the correct answer is:

Option: 7