If x=a and y=b then ……….. - Vidyadip

MCQ

If $x=a \tan \theta$ and $y=b \sec \theta$ then ………..

✓
$\frac{y^2}{ b ^2}-\frac{x^2}{ a ^2}=1$
B
$\frac{x^2}{ a ^2}-\frac{y^2}{ b ^2}=1$
C
$\frac{x^2}{ a ^2}+\frac{y^2}{ b ^2}=1$
D
$\frac{x^2}{ a ^2}-\frac{y^2}{ b ^2}=0$

✓

Answer

Correct option: A.

$\frac{y^2}{ b ^2}-\frac{x^2}{ a ^2}=1$

$2 a$
Explanation;
Hint:
$b\left(a^2-1\right)=(\sec \theta+\operatorname{cosec} \theta)\left[(\sin \theta+\cos \theta)^2-1\right] $
$=\frac{1}{\cos \theta}+\frac{1}{\sin \theta}\left[\sin ^2 \theta+\cos ^2 \theta+2 \sin \theta \cos \theta-1\right] $
$=\left[\frac{\sin \theta+\cos \theta}{\sin \theta \cos \theta}\right][1+2 \sin \theta \cos \theta-1] $
$=\left[\frac{\sin \theta+\cos \theta}{\sin \theta \cos \theta}\right] \times 2 \sin \theta \cos \theta$
$=2(\sin \theta+\cos \theta) $
$=2 a$

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