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Tangents and Normals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsTangents and Normals5 Marks
Question
Prove that $\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2$ touches the straight line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$ for all n $\in$ N, at all the point (a, b).
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Answer

The equation are
$\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}+\Big(\frac{\text{y}}{\text{b}}\Big)^\text{n}=2\ ...(1)$
$\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2\ ...(2)$
p = (a, b)
we need to prove (2) is the tangent to (1)
Differentiating (1) with respect to x, we get
$\text{n}\Big(\frac{\text{x}}{\text{a}}\Big)^\text{n}\times\frac{1}{\text{a}}+\text{n}\Big(\frac{\text{y}}{\text{b}}\Big)^{\text{n}-1}\times\frac{1}{\text{b}}\times\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\frac{\text{x}^{\text{n}-1}}{\text{a}^\text{n}}+\frac{\text{y}^{\text{n}-1}}{\text{b}^\text{n}}\times\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\Big(\frac{\text{x}}{\text{y}}\Big)^{\text{n}-1}\times\Big(\frac{\text{b}}{\text{a}}\Big)^\text{n}$
$\therefore$ slope $\text{m}=\Big(\frac{\text{dy}}{\text{dx}}\Big)_\text{p}=-\Big(\frac{\text{a}}{\text{b}}\Big)^{\text{n}-1}\times\Big(\frac{\text{b}}{\text{a}}\Big)^\text{n}$
$=-\frac{\text{b}}{\text{a}}$
Thus, the equation of tangent is
$(\text{y}-\text{b})=-\frac{\text{b}}{\text{a}}(\text{x}-\text{a})$
$\Rightarrow\text{bx}+\text{ay}=\text{ab}+\text{ab}$
$\Rightarrow\text{bx}+\text{ay}=2\text{ab}$
$\Rightarrow\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=2$

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