The condition that the line $x (\cos α) (y \sin α) = p$ may touch the curve ${(\frac{x}{a})}^{m} {(\frac{y}{b})}^{m} = 1$ is

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$a \cos (α) + b \sin (α) = 1$

$a \cos (α) + b \sin (α) = p$

${a \cos (α)}^{\frac{m}{m - 1}} b \sin {(α)}^{\frac{m}{m - 1}} = p^{\frac{m}{m - 1}}$

$a \cos (α) = 1$

answer is D.

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Detailed Solution

Given equation of curve is ${(\frac{x}{a})}^{m} {(\frac{y}{b})}^{m} = 1$

Differentiating the equation of curve w.r.t. $x$ , we get

${(\frac{x}{a})}^{m - 1} \cdot \frac{1}{a} + m {(\frac{y}{b})}^{m - 1} \cdot \frac{1}{b} \cdot \frac{d y}{d x} = 0$

On simplifying, we get $\frac{d y}{d x} = \frac{- b^{m} x^{m - 1}}{a^{m} y^{m - 1}}$

Now, at any point $P (x_{1}, y_{1})$ on the curve,

slope of tangent $= {(\frac{d y}{d x})}_{(x_{1}, y_{1})} = \frac{- b^{m} x_{1}^{m - 1}}{a^{m} y_{1}^{m - 1}}$

$∴$ Equation of tangent at $P$ is, $y - y_{1} = \frac{- b^{m} x_{1}^{m - 1}}{a^{m} y_{1}^{m - 1}} (x - x_{1})$

$\Rightarrow \frac{y y_{1}^{m - 1}}{b^{m}} - \frac{y_{1}^{m}}{b^{m}} = - \frac{x x_{1}^{m - 1}}{a^{m}} \frac{x_{1}^{m}}{a^{m}}$

i.e. $\frac{x}{a} {(\frac{x_{1}}{a})}^{m - 1} \frac{y}{b} {(\frac{y_{1}}{b})}^{m - 1} = {(\frac{x_{1}}{a})}^{m} {(\frac{y_{1}}{b})}^{m} = 1$

Since, $P$ lies on the curve, therefore the equation of tangent at $P (x_{1}, y_{1})$ on the curve is,

$\frac{\cos α}{\frac{1}{a} {(\frac{x_{1}}{a})}^{m - 1}} = \frac{\sin α}{\frac{1}{b} {(\frac{y_{1}}{b})}^{m - 1}} = 1$

Also, we have $x \cos α y \sin α = p$

$\frac{x}{a} {(\frac{x_{1}}{a})}^{m - 1} + \frac{y}{b} {(\frac{y_{1}}{b})}^{m - 1} {(\frac{a \cos α}{p})}^{\frac{m}{m - 1}} + {(\frac{b \sin α}{p})}^{\frac{m}{m - 1}} = 1$

This gives $\frac{x_{1}}{a} = {(\frac{a \cos α}{p})}^{\frac{1}{m - 1}}, \frac{y_{1}}{b} = {(\frac{b \sin α}{p})}^{\frac{1}{m - 1}}$

Since, point $P (x_{1}, y_{1})$ lies on the curve.

Therefore, we have ${(\frac{x_{1}}{a})}^{m} {(\frac{y_{1}}{b})}^{m} = 1$

$\Rightarrow a \cos {(α)}^{\frac{m}{m - 1}} b \cos {(α)}^{\frac{m}{m - 1}} = p^{\frac{m}{m - 1}}$

hence, the required condition is for the line touching the above curve is $(a \cos α)^{\frac{m}{m - 1}} (b \sin α)^{\frac{m}{m - 1}} = p^{\frac{m}{m - 1}}$