Exams › JEE Advanced › Maths
For a curve passing through the point (1, 1), the x-intercept of the tangent drawn at any point on the curve equals the y-coordinate (ordinate) of that point. Find the equation of the curve.
- x * e^(y/x) = e
- x * e^(x/y) = e
- y * e^(y/x) = e
- y * e^(x/y) = e
Correct answer: y * e^(x/y) = e
Solution
The tangent at point (x, y) has slope dy/dx. Its equation is Y - y = (dy/dx)(X - x). Setting Y = 0 gives X = x - y*(dx/dy) = x - y/(dy/dx). The condition says this x-intercept equals y (the ordinate), so x - y/(dy/dx) = y, giving dy/dx = y/(x - y). This is a homogeneous ODE. Substituting y = vx leads to the solution x*e^(y/x) = e after applying the initial condition (1,1).
Related JEE Advanced Maths questions
- Consider the differential equation associated with y = Σ (from i=1 to 3) C_i e^(m_i x), where C_i represents arbitrary constants and m₁, m₂, m₃ are the solutions of m³ - 7m + 6 = 0. If the equation is expressed as d³y/dx³ - 7 dy/dx + k y = 0, determine the value of k.
- Determine the order of the highest derivative raised to a power in the equation: (dy/dx)⁴ - 2x (d³y/dx³)² x² d²y/dx² d³y/dx³.
- The provided differential equation is expressed as a polynomial involving derivatives such as d²y/dx² and d³y/dx³. Among these, d³y/dx³ is the derivative of the highest order, and its greatest power in the equation is 2. What is the degree of this differential equation?
- When solving the differential equation d²y/dx² = 6x - 4, the integration gives dy/dx = 3x² - 4x + A. If dy/dx becomes zero at x = 1, what is the value of A?
- A curve passes through the point (1, π/6). Let the slope of the curve at each point (x, y) be y/x + sec(y/x), x > 0. Then the equation of the curve is -
- Consider y(x) as a solution to the differential equation (1 + e^x) dy/dx + y e^x = 1, with the initial condition y(0) = 2. Which of the following assertions is/are accurate?
⚔️ Practice JEE Advanced Maths free + battle 1v1 →