The curve $\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2, n \in N$ touches the line at the point…
The curve $\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2, n \in N$ touches the line at the point $(a, b)$, then the equation of the line is
$\frac{x}{a}-\frac{y}{b}=2$
$\frac{x}{a}+\frac{y}{2 b}=1$
$\frac{x}{a}+\frac{y}{b}=1$
$\frac{x}{a}+\frac{y}{b}=2$
Solution
Slope of the line $=\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $(a, b)$
$\begin{aligned} & =\frac{-n\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a}}{n \cdot\left(\frac{y}{b}\right)^{n-1} \cdot \frac{1}{b}} \text { at }(a, b) \\ & =\frac{-b}{a}\end{aligned}$
now equation of the straight line
$\begin{aligned} & y-b=-\frac{b}{a}(x-a) \\ & \Rightarrow a y-a b=-b x+a b \\ & \Rightarrow b x+a y=2 a b \\ & \Rightarrow \frac{x}{a}+\frac{y}{b}=2\end{aligned}$