Question

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

Answer 1

Let P(x, y) be any point on the curve. Then slope of the tangent at P is $\frac{dy}{dx}$.
It is given that the slope of the tangent at P(x,y) is equal to the ordinate i.e y.
Therefore $\frac{dy}{dx}$ = y
$$\begin{gathered} \Rightarrow \frac{1}{y}dy=dx \\ \Rightarrow \log y=x+\log C \\ \Rightarrow logy=\log e^x+logC \\ \Rightarrow y=C{e}^x \end{gathered}$$
Since, the curve passes through (1,1). Therefore, x=1 and y=1 .
Putting these values in equation obtained above we get,
$$\begin{gathered} 1=C{e}^1 \\ \Rightarrow C=\frac{1}{e} \\ puttingthesevaluesintheequationweget, \\ y=e^{x-1} \end{gathered}$$