Chapter 2.3, Problem 2E

To determine

To explain : The focus of a parabola with vertex $\left(0,0\right)$ and directrix $y=5$ .

Answer to Problem 2E

The required focus of the parabola is $\left(x,y\right)=\left(0,-5\right)$ .

Explanation of Solution

Given information : Vertex of the parabola is $\left(0,0\right)$ and equation of directrix is $y=5$ .

Since, a parabola is set of all points in a plane equidistant from a fixed point called the focus and a fixed line called the directrix.

Let us assume that $\left(x,y\right)$ be the focus of parabola.

Also, focus and vertex lies on a straight line and directrix also intersect this straight line at $\left(0,5\right)$ .

Now, from definition of parabola, vertex $\left(0,0\right)$ is midpoint of focus $\left(x,y\right)$ and $\left(0,5\right)$ .

Therefore, $\left(\frac{x+0}{2},\frac{y+5}{2}\right)=\left(0,0\right)$ .

Equating the above relation, we have

  $\frac{x+0}{2}=0\text{ and }\frac{y+5}{2}=0$

Therefore, $x=0$ and $y=-5$ .

Thus, required focus of the parabola is $\left(x,y\right)=\left(0,-5\right)$ .