Question

The $x$-intercept of the tangent to a curve is equal to the ordinate of the point of contact. The equation of the curve through the point $(1,1)$ is

• A. $x^{\frac{x}{y}}=e$
• B. $x{e}^{\frac{y}{x}}=e$
• C. $y{e}^{\frac{y}{x}}=e$
• D. $y{e}^{\frac{x}{y}}=e$

Answer 1

Answer: (D)

$Y-y=m(X-x)$
$\text{ for }X\text{-intercept }Y=0$
$X=x-\frac{y}{m}$
$\text{ hence }x-\frac{y}{m}=y$
or $\frac{dy}{dx}=\frac{y}{x-y}$
$xdy-ydy=ydx\Rightarrow -ydy=ydx-xdy$
$\frac{-ydy}{y^2}=\frac{ydx-xdy}{y^2}=$
$\Rightarrow \frac{-dy}{y}=d\left(\frac{x}{y}\right)$
Hence $-\ln y=\frac{x}{y}+C$
At $x=1,y=1\Rightarrow C=1$
Hence$y=e\cdot e^{-x/y}$
$e^{-x/y}=\frac{e}{y}$
$y{e}^{x/y}=e$