## Calculus (3rd Edition)

This is a parabola where $F=\left(0,c\right)$ is the focus and the line $y=-c$ is the directrix.
From Figure 23 we see that the line $y = nc$, where $n = 0,1,2,3,...$ intersects the circles ${x^2} + {\left( {y - c} \right)^2} = {c^2}{\left( {n + 1} \right)^2}$. For instance, $\begin{array}{*{20}{c}} n&{Line}&{Circle}\\ 0&{y = 0}&{{x^2} + {{\left( {y - c} \right)}^2} = {{\left( c \right)}^2}}\\ 1&{y = c}&{{x^2} + {{\left( {y - c} \right)}^2} = {{\left( {2c} \right)}^2}}\\ 2&{y = 2c}&{{x^2} + {{\left( {y - c} \right)}^2} = {{\left( {3c} \right)}^2}}\\ {...}&{...}&{...}\\ n&{y = nc}&{{x^2} + {{\left( {y - c} \right)}^2} = {c^2}{{\left( {n + 1} \right)}^2}} \end{array}$ From the figure attached we see that the dots are equidistant from the point $F=\left(0,c\right)$ and from the line $y=-c$. Thus, by definition this is a parabola where $F=\left(0,c\right)$ is the focus and the line $y=-c$ is the directrix.