Q105 / 160 · AP EAMCET 2022 (08 Jul Shift 1) · Mathematics
The sum of the intercepts made by a tangent drawn to the curve $\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$ at $(a, b)$ on the coordinate axes is
- A$a+b$
- B$a^2+b^2$
- C$2(a-b)$
- Correct answerD$2(a+b)$
Solution
Curve it :- $\left(\frac{\mathrm{x}}{\mathrm{a}}\right)^{\mathrm{n}}+\left(\frac{\mathrm{y}}{\mathrm{b}}\right)^{\mathrm{n}}=2$
Let we assume that $\mathrm{a}=1, \mathrm{~b}=2$
$\therefore$ Eqn of curve is :- $\mathrm{x}^{\mathrm{n}}+\left(\frac{\mathrm{y}}{2}\right)^{\mathrm{n}}=2$
$\begin{aligned} & \text { Now } \frac{d y}{d x}=\frac{-2^n x^{n-1}}{y^{n-1}} \\ & \text { At }(a, b)=(1,2) \\ & \left(\frac{d y}{d x}\right)_{(1,2)}=\frac{-2^n \times a^{n-1}}{b^{n-1}}=\frac{\left(-2^n\right) \times 1}{2^{n-1}}=-2^{n-n+1}=-2\end{aligned}$
$\therefore$ Eqn of tangent of the curve is:-
$\Rightarrow y-2=-2 x+2 \Rightarrow 2 x+y=4 \Rightarrow \frac{x}{2}+\frac{y}{4}=1$
$\therefore$ sum of interupt is :- $2+4=6$
Check by options to put $\mathrm{a}=1, \mathrm{~b}=2$ options $\mathrm{d}$ is satisfied.
Hence, the correct answer is option: 'D'.