Question

If $x=a{\cos }^2\theta \sin \theta$ and $y=a{\sin }^2\theta \cos \theta$, then $\frac{(x^2+y^2{)}^3}{x^2{y}^2}=$

• A. $a$
• B. $a^3$
• C. $a^2$
• D. $a^5$

Answer 1

The correct option is C $a^2$

Given : $x=a{\cos }^2\theta \sin \theta$ and $y=a{\sin }^2\theta \cos \theta$

$\frac{(x^2+y^2{)}^3}{x^2{y}^2}=\frac{(a^2{\cos }^4\theta {\sin }^2\theta +a^2{\cos }^2\theta {\sin }^4\theta {)}^3}{a^4{\cos }^6\theta {\sin }^6\theta }$

$=\frac{a^6{\cos }^6\theta {\sin }^6\theta {({\cos }^2\theta +{\sin }^2\theta )}^3}{a^4{\cos }^6\theta {\sin }^6\theta }$ ($\because si{n}^2\theta +co{s}^2\theta =1)$

$=a^2(1{)}^3=a^2$

Hence, option C is correct.