Question

Let tangent at a point P on the curve $x^{2m}{y}^{\frac{n}{2}}=a^{\frac{4m+n}{2}}$ meets the x-axis and y-axis at A and B respectively if AP : PB is $\frac{n}{\lambda m}$ where P lies between A and B then find the value of $\lambda$.

Answer 1

$x^{2m}{y}^{\frac{n}{2}}=a^{{}^{\frac{4m+n}{2}}}$

$\Rightarrow 2m\phi nx+\frac{n}{2}\phi ny=\frac{4m+n}{2}\phi na$

$\Rightarrow \frac{2m}{x}+\left(\frac{n}{2}\right)\frac{1}{y}\frac{dy}{dx}=0$

$\Rightarrow \frac{dy}{dx}=\left(\frac{-2m}{x}\right)\frac{2y}{n}$

Let P be the point $(x_1,y_1)$
So equation of tangent is $y-y_1=(\frac{4m}{n},\frac{y_1}{x_1})(x-x_1)$
Let tangent meet the axes the axes at A and B respectively
$A=(\frac{4m+m}{4m}{x}_0,0)$ $B=(0,\frac{4m+n}{n}.y_1)$
Let P divides AB in ratio $\mu :1$
$\therefore x_1=\frac{\mu 0.+1.\left(\frac{4m+n}{4m}\right)x_1}{\mu +1}\therefore 4m+n=4m(\mu +1)$
Also $\mu (4m+n)=n(\mu +1)\therefore \mu =\frac{n}{4m}$
$\therefore \lambda =4$