Question

Let '$a$' and '$b$' are positive number. If $(x,y)$ is a point on the curve $a{x}^2+b{y}^2=ab$ then the largest possible value of $xy$ is

• A. $\frac{\sqrt{ab}}{2}$
• B. $\sqrt{ab}$
• C. $\frac{ab}{a+b}$
• D. $\frac{2ab}{a+b}$

Answer 1

The correct option is A $\frac{\sqrt{ab}}{2}$

The point (x,y) on the curve can be written in polar coordinates as

$x=\sqrt{b}$cos$\theta$ and $y=\sqrt{a}$sin$\theta$

Thus,

$(xy{)}_{max}=(\sqrt{ab}$sin$\theta$cos$\theta {)}_{max}$

$=(\sqrt{ab}\frac{sin2\theta }{2}{)}_{max}$

$=\frac{\sqrt{ab}}{2}\because ($sin$2\theta {)}_{max}=1$

$\therefore$ Ans. is option A.