Question

The equations of tangent to the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2at,(a,b)$ is $5$.

• A. $\frac{x}{a}+\frac{y}{b}=2$
• B. $\frac{x}{a}+\frac{y}{b}=\frac{1}{2}$
• C. $\frac{x}{b}-\frac{y}{a}=2$
• D. $ax+by=2$

Answer 1

The correct option is A $\frac{x}{a}+\frac{y}{b}=2$

${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2at$

$n{\left(\frac{x}{a}\right)}^{n-1}\frac{1}{a}+n{\left(\frac{y}{b}\right)}^{n-1}\frac{y\prime }{b}=0$

so, $\frac{y\prime }{b}+\frac{1}{a}=0$ at $(a,b)$

so, $y\prime =\frac{-b}{a}$

so, tangent equation

$(y-b)=\frac{-b}{a}(x-a)$

$ay-ab=-bx+ab$

$bx+ay=2ab$

$\frac{x}{a}+\frac{y}{b}=2$