If the length of perpendicular from origin to any normal to the curve $y = f(x)$ is equal to its $y$ intercept, then
Step-by-Step Solution
Key Concept: For a normal to curve y = f(x) at point (x,y), the perpendicular distance from origin equals |y + x(dx/dy)|/√(1 + (dx/dy)²). Setting this equal to the y-intercept of the normal line and simplifying leads to the condition (dx/dy)² = 0, implying dx/dy = 0, which means y is constant.
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