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  • Let  be the set of all the natural numbers, for which the line <img alt="\frac{x}{a}+\frac{y}{b}=2" src="https://entrance

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}{\mathrm{a}}\right)^{\mathrm{n}}+\left(\frac{y}{\mathrm{~b}}\right)^{\mathrm{n}}=2 at the point (\mathrm{a}, \mathrm{b}), \mathrm{ab} \neq 0. Then:

Option: 1

\mathrm{S}=\phi


Option: 2

\mathrm{n}(\mathrm{S})=1


Option: 3

S=\{2 k: k \in \mathbf{N}\}


Option: 4

\mathrm{S}=\mathbf{N}


Answers (1)

best_answer

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

\mathrm{Slope\: of \: tangent \: at (a, b)}\\

\mathrm{n \cdot\left(\frac{x}{a}\right)^{n-1} \cdot \frac{1}{a} + n\left(\frac{y}{b}\right)^{n-1} \frac{1}{b} \frac{dy}{dx}=0 }

\mathrm{\left.\frac{d y}{d x}\right|_{(a, b)}=-\frac{b}{a}}

\therefore  Equation of tangent

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

\mathrm{\frac{x}{a}+\frac{y}{b}=2 \quad: \forall \quad n \in N}

Hence the correct answer is option 4

Posted by

Ritika Harsh

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