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  4. The equation of tangent to the curve ((x/a))n + ((y/b))n = 2 at the point (a,b) is

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

TS EAMCET 2017

Solution:

We have,
$\left(\frac{x}{a}\right)^{n}+\left(\frac{y}{b}\right)^{n}=2$
On differentiating w.r.t. $X$, we get
$\frac{n x^{n-1}}{a^{n}}+\frac{n y^{n-1}}{b^{n}} \frac{d y}{d x}=0$
$\Rightarrow \frac{d y}{d x}=\frac{-b^{n} \,x^{n-1}}{a^{n} \,y^{n-1}}$
$\Rightarrow \left(\frac{d y}{d x}\right)_{(a, b)}=\frac{-b^{n} \,a^{n-1}}{a^{n} \,b^{n-1}}=\frac{-b}{a}$
Equations of tangent at $(a, b)$ is
$y-b=\frac{-b}{a}(x-a)$
$\Rightarrow a y-a b=-b x+a b$
$\Rightarrow b x+a y=2\, a b$
$\Rightarrow \frac{b x}{a b}+\frac{a y}{a b}=\frac{2 \,a b}{a b}$
$\Rightarrow \frac{x}{a}+\frac{y}{b}=2$