If the point ( 7 3 3 ) lies on the curve traced by the mid-points of the line segments of the lines x cos + y sin = 7 , ( 0 2 ) between the coordinate axes, then is equal to [2023]

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Solution

Q.
If the point $(α, \frac{7 \sqrt{3}}{3})$ 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 coordinate axes, then $α$ is equal to [2023]

1 - 7

2 $- 7 \sqrt{3}$

3 $7 \sqrt{3}$

4 7

Ans.
(4)

$Given line is x \cos θ + y \sin θ = 7$

$x -intercept = \frac{7}{\cos θ}, y -intercept = \frac{7}{\sin θ}$

Let locus of intercept be $(h, k)$ .

$(h, k) = (\frac{7}{2 \cos θ}, \frac{7}{2 \sin θ})$

$h = \frac{7}{2 \cos θ}, k = \frac{7}{2 \sin θ}$

$α = \frac{7}{2 \cos θ}, \frac{7 \sqrt{3}}{3} = \frac{7}{2 \sin θ}$

$\cos θ = \frac{7}{2 α}, \sin θ = \frac{\sqrt{3}}{2}$

$\cos^{2} θ + \sin^{2} θ = 1$

$\Rightarrow \frac{49}{4 α^{2}} + \frac{3}{4} = 1 \Rightarrow \frac{49}{4 α^{2}} = \frac{1}{4}$

$α^{2} = 49$

$∴$ $α = 7 (As θ \in (0, \frac{π}{2}))$

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