Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
If the length of perpendicular from origin to any normal to the curve $y = f(x)$ is equal to its $y$ intercept, then
Curve is $x^2 = 4y + c$
Curve is $y = c$
Curve is $x = c$
Equation of normal to the curve $y = k$
Step-by-Step Solution
Key Concept: The perpendicular distance formula from a point to a line uses the normal form with derivative relationships.
The normal line at a point has equation $Y - y + \frac{dx}{dy}(X - x) = 0$. The perpendicular distance from the origin to this normal is $\frac{|y + x\frac{dx}{dy}|}{\sqrt{1 + (\frac{dx}{dy})^2}} = y + x\frac{dx}{dy}$, which simplifies to $1 + (\frac{dx}{dy})^2 = 1$.
Correct Answer: 3,4