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

• A. $\frac{1}{2}$
• B. 1
• C. $\frac{3}{2}$
• D. 2

Answer 1

The correct option is B

1

$\therefore x^{m}y=b^m$
Taking logarithm
$mlo{g}_{e}x+lo{g}_{e}y=mlo{g}_{e}b\therefore \frac{m}{x}+\frac{1}{y}\frac{dy}{dx}=0\therefore \frac{dy}{dx}=-\frac{my}{x}$
Equation of tangent at (x,y) is
$Y-y=-\frac{my}{x}(X-x)xY-xy=-myX+mxy\Rightarrow myX=xY=xy(1+m)\Rightarrow \frac{\frac{X}{x(1+m)}}{m}+\frac{x}{x(1+m)}=1$
$\therefore$ Area of triangle OAB
$=\frac{1}{2}.OA.OB$

$=\frac{1}{2}\left|\frac{x(1+m)}{m}\right|\left|y\right(1+m\left)\right|=\frac{\left|xy\right|(1+m{)}^2}{2\left|m\right|}Form=1,=\frac{\left|xy\right|\left(4\right)}{2}=2\left|xy\right|(\because xy=b)=2\left|b\right|=$ constant