Question

The focus of a parabola is at the origin and the equation of the directrix is $x = 2$ . What are the coordinates of the vertex of the parabola?
A. (2,0)
B. (0,2)
C. (1,0)
D. (0,1)

Answer 1

Hint: Let’s say that the directrix of the parabola intersects its axis at a point. The midpoint lying between this point and the focus of the parabola is the vertex of the parabola. It is at equivalent distance from the focus and the directrix.

Complete step by step solution: 
It is given that the focus of the parabola is at origin.
Thus, the coordinates of the focus are (0,0) respectively.
It is also given that the directrix of the parabola is $x = 2$ .
Now, we will find the equation of the axis of the parabola.
The axis of the parabola is a line perpendicular to the directrix and passing through the focus and vertex of the parabola both.
Thus, the slope of the line perpendicular to the directrix, $x = 2$ is 0.
As the axis passes through the focus, the point (0,0) also lies on the axis.
Hence, using the point slope form, the equation of a line having slope 0 and passing through (0,0) is:
$y - 0 = 0(x - 0)$
Therefore, the equation of the axis is $y = 0$ , which is the x-axis respectively.
Now, the directrix $x = 2$ will be intersecting the x-axis at point (2,0).
This is the point at which the directrix will be intersecting the axis of the parabola.
Hence, the vertex will be the midpoint between (2,0) and the focus (0,0).
This midpoint is (1,0) which gives us the coordinates of the vertex.
Therefore, the correct option is C.

Note: It is easier to solve the questions involving conic sections by using their respective figures. In this question, it can easily be observed that the axis of the parabola is x-axis and the directrix is lying to the left of the focus. This gives us a fair idea that the given parabola is of the form of ${y^2} = 4ax$ , which makes the further calculations much easier.