GATE2019 IN: 26

Question

The curve y = f(x) is such that the tangent to the curve at every point (x,y) has a y-axis intercept c, given by c = -y. Then,f(x) is proportional to

• A. x$^{-1}$
• B. x$^{2}$
• C. x$^{3}$
• D. x$^{4}$

 

Answer 1

Let the slope of the tangent at (x, y) be: m = dy/dx

The tangent line at (x, y) is: Y - y = m(X - x)

To find its y-axis intercept, put X = 0: Y - y = m(0 - x) Y = y - mx

So the y-intercept is: c = y - mx

Given: c = -y

Hence: y - mx = -y 2y = mx

Since m = dy/dx: x(dy/dx) = 2y

Separate variables: dy/y = 2(dx/x)

Integrate: ln|y| = 2ln|x| + C y = C * x^2

Therefore, f(x) is proportional to: x^2