Chapter B, Problem 48E

To determine

To calculate: To find the focus and directrix of the parabola $y=x^2$ and sketch the diagram

Answer to Problem 48E

The focus of parabola is $\left(0,\frac{1}{4}\right)$ and directrix is $y=-\frac{1}{4}$

Explanation of Solution

Given information: Equation of parabola is $y=x^2$

Formula Used:

Parabola is a set of points in the plane that are equidistant from a fixed point $F$ called the focus and a fixed line called the directrix

Calculation:

Equation of parabola is given as

  $y=x^2$

Comparing the equation withthe standard equation of parabola $x^2=4by$ with focus $F\left(0,b\right)$ and directrix $y=-b$

  $4b=1$

On solving,

  $b=\frac{1}{4}$

Thus,

Focus of parabola is $\left(0,\frac{1}{4}\right)$ and directrix is $y=-\frac{1}{4}$

On sketching the parabola,

  

Conclusion:

Hence, focus of parabola is $\left(0,\frac{1}{4}\right)$ and directrix is $y=-\frac{1}{4}$