Question

The minimum distance from origin to any point on $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ is

• A. $a$
• B. $\frac{a}{2}$
• C. $2a$
• D. $a^{2/3}$

Answer 1

Answer: (A)

$a$

Let the point on curve be $(asi{n}^3\theta ,aco{s}^3\theta )$
So distance from origin $=\sqrt{a^{2}si{n}^6\theta +a^{2}co{s}^6\theta }$
$\frac{d\left(d\right)}{d\theta }=\frac{1}{2}(\frac{a^2(6si{n}^5\theta cos\theta -6co{s}^5\theta sin\theta )}{\sqrt{a^{2}si{n}^6\theta +a^{2}co{s}^6\theta }}$
$0=3a^{2}sin\theta cos\theta (si{n}^4\theta -co{s}^4\theta )$
$=si{n}^2\theta co{s}^2\theta$
$0=si{n}^4\theta$
$\theta =0,n\pi$
$d=a$