Normal to xy = c² at Parameter t

Normal to xy = c² at (ct, c/t). Slope = t². Up to four normals can be drawn from a general point to xy = c².

At $(ct, c/t)$, tangent slope $= -1/t^2$, so normal slope $= t^2$.

Normal through $(ct, c/t)$:

$$y - \frac{c}{t} = t^2(x - ct)$$ $$yt - c = t^3(xt - ct^2) \quad \text{(multiply by } t\text{)}$$

Wait — multiply through directly:

$$y - c/t = t^2 x - ct^3$$ $$y = t^2 x - ct^3 + c/t$$

Rearranging to standard form (multiply through by $t$):

$$yt = t^3x - ct^4 + c$$ $$xt^3 - yt = ct^4 - c = c(t^4-1)$$ $$xt^3 - yt = c(t^4-1)$$

Normals from a point $(h, k)$: Substituting:

$$ht^3 - kt = c(t^4-1)$$ $$ct^4 - ht^3 + kt - c = 0$$

This is a quartic in $t$. By Vieta's formulas for roots $t_1, t_2, t_3, t_4$:

$$t_1+t_2+t_3+t_4 = \frac{h}{c}$$ $$t_1t_2t_3t_4 = -1$$

The product of the four parameters at the feet of the normals from any point is $-1$.

Up to four distinct real normals can be drawn from a general exterior point to $xy = c^2$.