Mathematics
Mathematics
Problem
Consider the set of all points that are equidistant from the point and the line . (i) Show that the vertex of the curve lies at the origin. (ii) Find the Cartesian equation of the curve. (iii) State the coordinates of its focus, the equation of its directrix, and the length of its latus rectum. (iv) Sketch the curve on the Cartesian plane, clearly marking the focus, directrix and vertex.Intuitive explanation
This problem is about the definition of a parabola: the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The key insight is that distance to a point equals distance to a line, which gives you a powerful equation to work with. Once you set up that equation algebraically, the parabola's shape and all its properties—vertex, focus, directrix, and latus rectum—fall out naturally.
Intuitive explanation
This problem is about the definition of a parabola: the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). The key insight is that distance to a point equals distance to a line, which gives you a powerful equation to work with. Once you set up that equation algebraically, the parabola's shape and all its properties—vertex, focus, directrix, and latus rectum—fall out naturally.