Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
Tangent is drawn at the point $(x_i, y_i)$ on the curve $y = f(x)$, which intersects the x-axis at $(x_{i+1}, 0)$. Now, again a tangent is drawn at $(x_{i+1}, y_{i+1})$ on the curve which intersects the x-axis at $(x_{i+2}, 0)$ and the process is repeated $n$ times, i.e., $i = 1, 2, 3, ........., n$. If $x_1, x_2, x_3, ........., x_n$ form an arithmetic progression with common difference equal to $\log_2 e$ and curve passes through $(0, 2)$. Now if curve passes through the point $(-2, k)$, then the value of $k$ is ______.
Step-by-Step Solution
Key Concept: Tangent line geometry where successive x-intercepts form an arithmetic progression constrains the differential equation.
The tangent line at point $P(x_1, y_1)$ is $y - y_1 = \frac{dy}{dx}(x-x_1)$. This intersects the x-axis at $x_2 = x_1 - \frac{y_1}{dy/dx}$. Given that $x_1, x_2, x_3, \ldots$ form an AP with common difference $\log_2 e$, we have $x_2 - x_1 = -\frac{y_1}{dy/dx} = \log_2 e$. This gives $\frac{dy}{y}\log_2 e = -dx$. Integrating: $\log_2 y = -x\log_2 e + c$, so $y = ke^{-x\log_2 e}$. Using $x=-2$ gives $y = 2e^{2\log_2 e} = 2 \cdot 4 = 8$.
Correct Answer: 8