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. $xe ^{\frac{ y }{ x }}= e$
• 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 $\frac{d y}{d x}=\frac{y}{x-y}$
$x d y-y d y=y d x \Rightarrow -y d y=y d x-x d y$
$\frac{-y d y}{y^2}=\frac{y d x-x d y}{y^2}= $
$\Rightarrow \frac{-d y}{y}=d\left(\frac{x}{y}\right)$
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 $