Chapter 10.2, Problem 82PE

(a)

To determine

To prove: The distance between $F_1$ and $V_2$ is $\left(c+a\right)$ and the distance between $F_2$ and $V_2$ is $\left(c-a\right)$ and the difference between these distances is $2a$ . The given figure is:

(b)

To determine

To calculate: The expression that represents the difference of the distance from $F_1$ to $\left(x,y\right)$ and from $F_2$ to $\left(x,y\right)$ .

(c)

To determine

To prove: The equation $\sqrt{{\left(x+c\right)}^2+y^2}-\sqrt{{\left(x-c\right)}^2+y^2}=2a$ can be written as $cx-a^2=a\sqrt{{\left(x-c\right)}^2+y^2}$ .

(d)

To determine

To prove: The equation $cx-a^2=a\sqrt{{\left(x-c\right)}^2+y^2}$ can be written as $\left(c^2-a^2\right)x^2-a^2{y}^2=a^2\left(c^2-a^2\right)$ .

(e)

To determine

To prove: The equation $\left(c^2-a^2\right)x^2-a^2{y}^2=a^2\left(c^2-a^2\right)$ can be written as $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ .