First slide

Equation of line in Straight lines

Question

If $P$ and $Q$ are the lengths of the perpendiculars from the origin to the lines $x \cos θ + y \sin θ = k \cos 2 θ$ and $x s e c θ - y \cos e c θ = k$ respectively then

Moderate

A

$p^{2} - 4 q^{2} = k^{2}$
B

$p^{2} + 4 q^{2} = k^{2}$
C

$4 p^{2} + 4 q^{2} = k^{2}$
D

$p^{2} + q = k^{2}$

Solution

$\begin{array}{l} p = \frac{| k \cos 2 θ |}{\sqrt{\cos^{2} θ + \sin^{2} θ}} = | k \cos 2 θ | \\ q = \frac{| k |}{\sqrt{\sec^{2} θ + {coses}^{2} θ}} = | k \sin θ \cos θ | \\ = | k (1 / 2) \sin 2 θ | \end{array}$
So $p^{2} + 4 q^{2} = k^{2} (\cos^{2} 2 θ + \sin^{2} 2 θ) = k^{2} .$

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