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

Answer 1

The correct option is A $y{e}^{\frac{x}{y}}=C$

Let the equation of tangent at P(x, y) is
$(Y-y)=\frac{dy}{dx}(X-x)x-intercept=\frac{\frac{xdy}{dx}-y}{\frac{dy}{dx}}\therefore y=\frac{\frac{xdy}{dx}-y}{\frac{dy}{dx}}\Rightarrow \frac{dy}{dx}=\frac{y}{x-y}y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}\Rightarrow \frac{1-v}{v^2}dv=\frac{dx}{x}\Rightarrow logx+logv)=-\frac{1}{v}\Rightarrow Cy=e^{-\frac{x}{y}}\Rightarrow y\times e^{\frac{x}{y}}=C$