Question

Let tangent at a point P on the curve $x^{2m}{Y}^{\frac{n}{2}}=a^{\frac{4m+n}{2}}(m,n\in N,niseven)$, meets the x-axis and y-axis at A and B respectively, if $AP:PBis\frac{n}{\lambda m}$, where P lies between A and B, then find the value of $\lambda$

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

The correct option is B 4

Given curve is $x^{2m}{y}^{\frac{n}{2}}=a^{\frac{4m+n}{2}}$
applying log on both sides gives
$2m\log x+\frac{n}{2}\log y=\frac{4m+n}{2}\log a$
differentiating w.r.t $x$
$\frac{2m}{x}+\frac{n}{2y}\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{4m}{n}\frac{y}{x}$
Equation of tangent at $P(x_1,y_1)$ is
$y-y_1=-\frac{4m}{n}\frac{y_1}{x_1}(x-x_1)$
above equation meets $x$-axis at $A$ and $y$-axis at $B$ respectively
$A\left(\frac{4m+n}{4m}{x}_1,0\right)$
$B(0,\frac{4m+n}{n}{y}_1)$
given $AP:PB$ = $\frac{n}{\lambda m}$
$\Rightarrow \frac{\lambda m\left(\frac{4m+n}{4m}\right)x_1}{\lambda m+n}=x_1$ (by using section formula)
$\therefore$ $\lambda =4$
Hence, option B.