First slide

Equation of line in Straight lines

Question

The angle made by the perpendicular drawn from the origin to the line $x + 2 y - 2 \sqrt{5} = 0$ with $X -$ axis is

Moderate

A

$\tan^{- 1} (3)$
B

$\tan^{- 1} (1)$
C

$\tan^{- 1} (2)$
D

$\tan^{- 1} (4)$

Solution

$The given equation is x + 2 y - 2 \sqrt{5} = 0$
$To reduce the general equation of the line a x + b y + c = 0 in normal form, isolate the constant$
$term and then divide both sides with \sqrt{a^{2} + b^{2}} and make constant term must be positive.$
Hence,
$x + 2 y = 2 \sqrt{5}$
$Divide both sides with \sqrt{1^{2} + 2^{2}} = \sqrt{5}$
$\frac{x}{\sqrt{5}} + \frac{2 y}{\sqrt{5}} = 2$
$By comparing the above equation with normal form of the line x \cos α + y \sin α = p$
It implies
$\cos α = \frac{1}{\sqrt{5}}, \sin α = \frac{2}{\sqrt{5}}$
$Hence \tan α = 2$
$Therefore, the angle required is α = \tan^{- 1} (2)$

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