If the point α , 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 ∈ 0 , π 2 between the co-ordinates axes, then α is equal to

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TestHub · Accepted Answer

The straight line is shown below with the intercepts.

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

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

$∵ θ \in (0, \frac{π}{2})$

$\Rightarrow \frac{49}{4 h^{2}} + \frac{49}{4 k^{2}} = 1 ∴ x$ and $y$ are positive

$∴$ Locus $= \frac{1}{x^{2}} + \frac{1}{y^{2}} = \frac{4}{49}$

Now $(α, \frac{7 \sqrt{3}}{3})$ lies on curve

$\Rightarrow \frac{1}{α^{2}} + \frac{9}{147} = \frac{4}{49}$

$\frac{1}{α^{2}} = \frac{1}{49}$

$∴ α = \pm 7$

$∴ α = 7$

Hence this is the required option.

Mathematics - Straight Lines Question with Solution | TestHub

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