Tangents and Normals of Conics

Try this

1. Drag $Q$ toward $P$. The slope of the secant (solid line) changes. Which number is it chasing? When $Q$ runs into $P$, what is the angle between the two lines?
2. Switch to the parabola. Through $P$, draw the horizontal line parallel to the axis of symmetry. How many points does it share with the parabola? Is it a tangent?
3. Switch to “Reflection” and drag $P$. How are the two readings related? Try it once for each of the three curves.

For a circle, “a line with exactly one common point with the circle” is a tangent. For conics this statement is no longer reliable, so we use another definition:

Through a point $P$ of the curve draw a secant $PQ$ and let $Q$ approach $P$ along the curve. The limiting position of the secant is the tangent to the curve at $P$. The line through $P$ perpendicular to the tangent is the normal.

Why: from secant to tangent

Take as an example the point $P(x_0,y_0)$ on the parabola $y^2=4x$ ($p=2$); in the figure $y_0=1.5$.

It also works for the general second-degree curve $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$: $x^2\to x_{0}x$, $y^2\to y_{0}y$, $xy\to \frac{x_{0}y+x{y}_0}{2}$, $x\to \frac{x+x_0}{2}$, $y\to \frac{y+y_0}{2}$.

To find a tangent with a given slope $k$, let it be $y=kx+m$, substitute, and set the discriminant to $0$. For the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ the result is $m^2=a^2{k}^2+b^2$.

The reflection property

• Ellipse: the normal bisects $\angle F_{1}P{F}_2$. Light leaving one focus passes through the other focus after reflecting off the ellipse.
• Hyperbola: the tangent bisects $\angle F_{1}P{F}_2$. Light aimed at one focus reflects as if it came from the other focus.
• Parabola: light coming in parallel to the axis reflects into the focus; conversely, a light source at the focus reflects into a parallel beam. Searchlights and satellite dishes use this property.

Take the ellipse: the definition is all you need for the reason. Pick any point $T\ne P$ on the tangent. $T$ is outside the ellipse, so $|T{F}_1|+|T{F}_2|>2a=|P{F}_1|+|P{F}_2|$. Hence $P$ is the point of the tangent with the smallest “sum of distances to the two foci”. Reflect $F_2$ in the tangent to get $F_2^{\prime }$; the shortest path $F_{1}P{F}_2^{\prime }$ is a straight line, so $P{F}_1$ and $P{F}_2$ make equal angles with the tangent, which is equivalent to the normal bisecting $\angle F_{1}P{F}_2$.

Practice