The equation of the directrix of the parabola with vertex at the origin and having the axis along the $x$-axis and a common tangent of slope $2$ with the circle $x^2 + y^2 = 5$ is/are :
Step-by-Step Solution
Key Concept: A line is tangent to a circle when the distance from the center to the line equals the radius; for a parabola, use the condition that the discriminant of the substitution is zero.
The line $y = 2x + c$ is tangent to the circle $x^2 + y^2 = 5$ when $c^2 = 25$, so $c = \pm 5$. For a parabola $y^2 = 4ax$ with the same tangent line, substituting gives $a = \pm 10$. Thus, the parabola equations are $y^2 = \pm 40x$ with directrices $x = \mp 10$.
Correct Answer: 1,3