If the point $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ lies on the curve traced by the mid-points of the line segments of the lines x $\cos \theta+\mathrm{y} \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$ between the coordinates axes, then $\alpha$ is equal to

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Straight Lines

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

cos θ + y sin θ = 7, θ \in (0, \frac{π}{2})

between the coordinates axes, then

α

is equal to

(A)7✓ Correct
(B)-7
(C) $- 73$
(D) $73$

Answer: (A)

Solution

The line

x cos θ + y sin θ = 7

has x-intercept

\frac{7}{c o s θ}

and y-intercept

\frac{7}{s i n θ}

. The midpoint is

(\frac{7}{2 c o s θ}, \frac{7}{2 s i n θ})

. For the point

(α, \frac{7 3}{3})

to lie on this locus, we have

\frac{7}{2 s i n θ} = \frac{7 3}{3}

, giving

sin θ = \frac{3}{2 3} = \frac{3}{2}

, so

θ = \frac{π}{3}

. Then

α = \frac{7}{2 c o s ( π /3 )} = \frac{7}{2 \cdot \frac{1}{2}} = 7

JEE Main · 2023 · 12 Apr 2023 · Shift 1 · Single correct option

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