Question

If the tangent at P on the curve $x^m{y}^n=a^{m+n}$ meets the co-ordinates axes at A and B, then $AP:PB=$

• A. $m^2:n^2$
• B. $m^3:n^3$
  
• C. $m:n$
  
• D. $2m:n$
  

Answer 1

The correct option is C $m:n$

$x^m{y}^n=a^{m+n}.$

Take $log$ both sides

$m\log x+n\log y=(m+n)\log a$

Differentiating w.r.t $x$, we get

$m\cdot \frac{1}{x}+n\cdot \frac{1}{y}\frac{dy}{dx}=0$

$\therefore \frac{dy}{dx}=-\frac{m}{n}\frac{y}{x}$

Hence the equation of the tangent is

$Y-y=-\frac{m}{n}\frac{y}{x}(X-x)$

$\Rightarrow myX+nxY=(m+n)\cdot xy$...(1)

Let the tangent (1) meet the axes in points $A$ and $B$.

Putting $Y=0,$ we get $A=[\frac{m+n}{m}x,0]$

Putting $X=0$ we get $B=(0,\frac{m+n}{m}y)$

Point of contact $P$ is $(x,y).$

Let $P$ divide $AB$ in the ratio $\lambda :1$

$\therefore x=\frac{\lambda .0+1\cdot \frac{m+n}{n}x}{\lambda +1}$$\Rightarrow (\lambda +1)x=(\frac{m}{n}+1)x$

$\Rightarrow \lambda +1=\frac{m}{n}+1$

$\therefore \lambda =\frac{m}{n}$