The value of m for which the area of the triangle included between the axes

Discussion Forum : Application Of Derivatives

Question -

The value of m for which the area of the triangle included between the axes and any tangent to the curve $x^{m} y = b^{m}$ is constant is

Options:

A .

\frac{1}{2}

B . 1

C .

\frac{3}{2}

D . 2

Answer: Option B
:
B

$∴ x^{m} y = b^{m}$
Taking logarithm
$m l o g_{e} x + l o g_{e} y = m l o g_{e} b ∴ \frac{m}{x} + \frac{1}{y} \frac{d y}{d x} = 0 ∴ \frac{d y}{d x} = - \frac{m y}{x}$
Equation of tangent at (x,y) is
$Y - y = - \frac{m y}{x} (X - x) x Y - x y = - m y X + m x y \Rightarrow m y X = x Y = x y (1 + m) \Rightarrow \frac{\frac{X}{x (1 + m)}}{m} + \frac{x}{x (1 + m)} = 1$
$∴$ Area of triangle OAB
$= \frac{1}{2} . O A . O B$

$= \frac{1}{2} ∣ ∣ \frac{x (1 + m)}{m} ∣ ∣ | y (1 + m) | = \frac{| x y | (1 + m)^{2}}{2 | m |} F o r m = 1, = \frac{| x y | (4)}{2} = 2 | x y | (∵ x y = b) = 2 | b | =$ constant

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