Answer to Question #103326 in Analytic Geometry for Sourav Mondal

Question

Give the equation of a conic which is symmetric to the line x+2 = 0.

Answer 1

The conic symmetric to line x+2=0 would be a parabola with the given line acting as its axis.

Let the vertex of this parabola lie on the x-axis itself. Hence, it is of the form $(a,0)$. Clearly, the vertex also lies on the axis as well.

$\implies a=-2 \implies (-2,0)$ is the vertex of the parabola.

Equation of a parabola with axis x=0 is $y=x^2$. Thus, by shifting origin to (-2,0), we get

$y=(x+2)^2$

$\implies y=x^2+4x+4$ is the required equation of the conic which is symmetric to x+2=0