First slide

Equation of line in Straight lines

Question

If $p_{1}, p_{2}$ denote the lengths of the perpendiculars from the origin on the lines $x s e c a + y \cos e c a = 2 a$ and $x \cos a + y \sin a = a \cos 2 a$ respectively,

Moderate

A

$4 \sin^{2} 4 α$
B

$4 \cos^{2} 4 α$
C

$4 \cos e c^{2} 4 α$
D

$4 s e c^{2} 4 α$

Solution

$\begin{array}{l} p_{1}^{2} + p_{2}^{2} = \frac{4 a^{2}}{\sec^{2} α + {cosec}^{2} α} + \frac{a^{2} \cos^{2} 2 α}{\cos^{2} α + \sin^{2} α} \\ = a^{2} (\sin^{2} 2 α + \cos^{2} 2 α) = a^{2} \end{array}$
and $p_{1}^{2} p_{2}^{2} = a^{4} \sin^{2} 2 α \cos^{2} 2 α = (1 / 4) a^{4} \sin^{2} 4 α$
$∴ {(\frac{p_{1}}{p_{2}} + \frac{p_{2}}{p_{1}})}^{2} = \frac{{(p_{1}^{2} + p_{2}^{2})}^{2}}{p_{1}^{2} p_{2}^{2}} = \frac{4}{\sin^{2} 4 α} = 4 {cosec}^{2} 4 α .$

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