Secants and tangents of a parabola
A secant of a parabola is a line, or line segment, that joins two distinct points on the parabola. A tangent is a line that touches the parabola at exactly one point. Finding tangents to curves is historically an important problem going back to Fermat, and is a key motivator for the differential calculus.
In this step we will see how to algebraically find both secants and tangents to a parabola. So this section gives an alternative algebraic perspective on calculus concepts!
A parametrization of the parabola
The standard parabola has a nice parametrization of the form
For any value of , this formula for gives us a point lying on the parabola.
In the figure we see the parabola . Note that as varies, the - coordinate of moves linearly, while the - coordinate moves quadratically.
Secant of a parabola
Now suppose that we have two points and on the parabola . The line joining and has slope
and so it has an equation of the form . To determine the -intercept , we may substitute in the coordinates . We get that so the equation of the secant is
Q1 (E): What is the equation of the secant through the points on the parabola whose - coordinates are and ?
Tangent to a parabola
What happens to our formula for the secant when and are very close? Better yet, what happens when we set , so that ? This is actually a very interesting situation that rewards careful examination. We are parachuting into the hallowed halls of calculus using algebra!
In this case the equation of the secant becomes the equation of the tangent to the parabola: starting with the equation of the secant and substituting gives:
Note that in particular the slope of the tangent line to the point with -coordinate is . This number is usually called the derivative of the function at the point , and in calculus courses is introduced following a study of limits. Our purely algebraic techniques allowed us to find the equation of the tangent to the parabola . Since any other parabola is just a translate of such, we see that we can deal with secants and tangents for general parabolas, without limits!
Q2 (E): What is the equation of the tangent to the parabola at the point ?
Two properties of the parabola
Here are two interesting challenges for you. Let’s see if we can derive some interesting aspects of the geometry of the parabola. The second one will require some thorough thinking. You will need to know that two lines are perpendicular precisely when their slopes multiply to .
Q3 (M): Consider the points and on the parabola . Show that the two tangents to the parabola at and meet at .
Notice that the - coordinate of is halfway between and , and does not depend on the particular parabola.
Q4 (C): Show that the two tangents to the parabola through the points and are perpendicular precisely when the line passes through the focus of the parabola.
A1. The secant is the line .
A2. Apply the formula with and to get the tangent: .
A3. Solve the equations and simultaneously.
A4. You can do it!
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