What is the directrix of the parabola?
Consider the following for the next two (02) items that follow: A parabola passes through (1, 2) and satisfies the differential equation \frac{dy}{dx}=\frac{2y}{x}, x \gt 0, y \gt 0.
- A. y=-\frac{1}{8} ✓
- B. y=\frac{1}{8}
- C. x=-\frac{1}{8}
- D. x=\frac{1}{8}
Correct Answer: A. y=-\frac{1}{8}
Explanation
Separating variables, \int \frac{dy}{y} = \int \frac{2dx}{x} \implies \ln y = 2\ln x + C \implies y = kx^2. Since it passes through (1,2), 2 = k(1)^2 \implies k = 2. The equation is x^2 = \frac{1}{2}y. Comparing with x^2 = 4ay, 4a = \frac{1}{2} \implies a = \frac{1}{8}. The directrix of x^2 = 4ay is y = -a, so y = -\frac{1}{8}.
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