Chapter 7.5, Problem 35WE

To determine

Prove that ( h − r , s ) is also on the parabola $y-k={\left(x-h\right)}^2$ .What fact is established about the parabola.

Answer to Problem 35WE

Parabola is symmetric.

Explanation of Solution

Given:

The point ( h + r , s ) is on the parabola $y-k={\left(x-h\right)}^2$ .

Calculation:

Since ( h + r , s ) is on the parabola $y-k={\left(x-h\right)}^2$ , it must satisfy the equation , hence:

  $\begin{array}{l}y-k={\left(x-h\right)}^2\\ \Rightarrow s-k={\left(h+r-h\right)}^2\\ \Rightarrow s-k=r^2\end{array}$

Consider the point ( h − r , s ) and substitute it in the equation $y-k={\left(x-h\right)}^2$ :

  $\begin{array}{l}y-k={\left(x-h\right)}^2\\ \Rightarrow s-k={\left(h-r-h\right)}^2\\ \Rightarrow s-k=r^2\end{array}$

So, the point ( h − r , s ) is also on the parabola $y-k={\left(x-h\right)}^2$ .

Since, ( h − r , s ) and ( h + r , s ) lie on the parabola , so , the point on one arm has mirror image on the other arm of the parabola. Hence, the parabola is symmetric.