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) \[ax+by=2\]
E) \[ax-by=2\]
Correct Answer: A
Solution :
The equation of curve is \[\frac{{{x}^{n}}}{{{a}^{n}}}+\frac{{{y}^{n}}}{{{b}^{n}}}=2\] On differentiating w. r. t.\[x,\]we get \[\frac{n{{x}^{n-1}}}{{{a}^{n}}}+\frac{n{{y}^{n-1}}}{{{b}^{n}}}\frac{dy}{dx}=0\] \[\Rightarrow \] \[\frac{dy}{dx}=-\frac{{{x}^{n-1}}.{{b}^{n}}}{{{a}^{n}}.{{y}^{n-1}}}\] \[\therefore \] \[{{\left( \frac{dy}{dx} \right)}_{(a,b)}}=-\frac{{{a}^{n-1}}.{{b}^{n}}}{{{a}^{n}}.{{b}^{n-1}}}=-\frac{b}{a}\] \[\therefore \]Equation of tangent to the curve at (a, b) is given by \[(y-b)=-\frac{b}{a}(x-a)\] \[\Rightarrow \] \[ay+bx=2ab\] \[\Rightarrow \] \[\frac{x}{a}+\frac{y}{b}=2\]
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