Chapter 2, Problem 3CT

To determine

To calculate: Focus, Directrix and Axis of Symmetry of Parabola.

Answer to Problem 3CT

Focus: $\left(\frac{1}{8},0\right)$

Directrix: $x=-\frac{1}{8}$

Axis of Symmetry: $y=0$

Explanation of Solution

Given information: Equation of Parabola Y $x=2y^2$

Calculation:

Compare the equation with vertex form:

13 $x=a{(y-k)}^2+h$

We get:

  $a=2,h=0,k=0$

The distance from the vertex to a focus of the parabola is given by

  $p=\frac{1}{4a}=\frac{1}{4.2}=\frac{1}{8}$

Since, $a$ is positive in our case, hence parabola opens right.

The focus is given by:

  $(h+p,k)=\left(0+\frac{1}{8},0\right)=\left(\frac{1}{8},0\right)$

The axis if symmetry is the line that passes through the focus and vertex and hence axis of symmetry is given by:

  $y=0$

The equation of directrix is given by formula:

  $\begin{array}{l}x=h-p\\ =>x=0-\frac{1}{8}\\ =>x=-\frac{1}{8}\end{array}$

Graph: