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 y = mx + c is tangent to circle x² + y² = r² when the distance from center to line equals r, giving |c|/√(m²+1) = r. For parabola y² = 4ax, substituting the tangent line and requiring a double root determines the parameter a, whose negative reciprocal gives the directrix position x = -1/a.
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