Chapter B, Problem 47E

To determine

To calculate: To show that the for a parabola, $x^2=-4py$

  $x^2=-4py$

Answer to Problem 47E

It is proved that $x^2=4py$

Explanation of Solution

Given information: $P\left(x,y\right)$ is any point on the parabola with focus $\left(0,p\right)$ and directrix $y=-p$

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:

Consider parabola with focus $F\left(0,p\right)$ and directrix $y=-p$

Let $P\left(x,y\right)$ is any point on the parabola

Distance between the point and focus is $FP$

  $\begin{array}{l}FP=\sqrt{{\left(x-0\right)}^2+{\left(y-p\right)}^2}\\ FP=\sqrt{x^2+y^2+p^2-2yp}\end{array}$

Let us assume that thedistance between the point and directrix $y=-p$ is $K$

  $\begin{array}{l}K=\frac{y+p}{\sqrt{0^2+1^2}}\\ K=\frac{y+p}{\sqrt{1^2}}\\ K=y+p\end{array}$

Now, the definition of parabola,

  $FP=K$

Thus,

  $\begin{array}{l}\sqrt{x^2+y^2+p^2-2p}y=y+p\\ {\left(\sqrt{x^2+y^2+p^2-2py}\right)}^2={\left(y+p\right)}^2\\ x^2+y^2+p^2-2py=y^2+p^2+2py\\ x^2-2py=2py\\ x^2=4py\end{array}$

Conclusion:

Hence, proved that $x^2=4py$