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 the point on the curve y = f(x) such that tangent at P cuts the coordinate axes at A and B.

It cuts the axes at A and B so, equation of tangent at P(x, y)

Given, intercept on x – axis = y

We can see that it is a linear differential equation.

Comparing it with \(\frac{dy}{dx}+Py = Q\)

Solution of the given equation is given by

As the equation passing through (1, 1)

0 = – 1 + c

⇒ c = 1

Putting the value of c in equation (1)

\(\therefore \frac{\text{x}}{\text{y}}=\) - log y + 1

⇒ x = y – ylogy