Angle between the lines x a +y b =1 and x a -y b =1 is - Vidyadip

MCQ

Angle between the lines $\frac{x}{ a }+\frac{y}{b}=1$ and $\frac{x}{ a }-\frac{y}{b}=1$ is

✓
$2 \tan ^{-1} \frac{b}{a}$
B
$\tan ^{-1} \frac{2 a b}{a^2+b^2}$
C
$\tan ^{-1} \frac{a^2-b^2}{a^2+b^2}$
D
None of these

✓

Answer

Correct option: A.

$2 \tan ^{-1} \frac{b}{a}$

(A)
The lines are $b x+ a y- ab =0$ and $b x- a y- ab =0$.
Hence, the required angle is
$\theta=\tan ^{-1}\left|\frac{\frac{- b }{ a }-\frac{ b }{ a }}{1-\frac{ b ^2}{ a ^2}}\right|$
$=\tan ^{-1}\left|\frac{2 a b}{b^2-a^2}\right|$
$=2 \tan ^{-1} \frac{b}{ a }$ ...$\ldots\left[\because 2 \tan ^{-1} \frac{y}{x}=\tan ^{-1}\left|\frac{2 x y}{y^2-x^2}\right|\right]$

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