Question

Let $S$ be the set of all the natural numbers, for which the line $\frac{x}{a}+\frac{y}{b}=2$ is a tangent to the curve $\left(\frac{ x }{ a }\right)^{ n }+\left(\frac{ y }{ b }\right)^{ n }=2$ at the point $(a, b), a b \neq 0$. Then:

• A. $S=\phi$
• B. $n ( S )=1$
• C. $S =\{2 k : k \in N \}$
• D. $S = N$

Answer 1

Answer: (D)

$\left(\frac{ x }{ a }\right)^{ n }+\left(\frac{ y }{ b }\right)^{ n }=2$
Slope of tangent at $(a, b)$
$n \cdot\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a}+n\left(\frac{x}{b}\right)^{n-1} \cdot \frac{1}{b} \frac{d y}{d x}=0$
$\left.\frac{ dy }{ dx }\right|_{(a, b)}=-\frac{ b }{ a }$
$\therefore$ Equation of tangent
$y-b=-\frac{b}{a}(x-a)$
$\frac{ x }{ a }+\frac{ y }{ b }=2 \forall n \in N$