Conic Sections: The Ellipse - Part 1

The ellipse is the type of conic section formed by the locus of a moving point P, such that the ratio (eccentricity) of its distance to a fixed focal point F, to a fixed imaginary line called the directrix D is a positive constant that is less than 1.

Try this experiment at home to better understand how mathematically, an ellipse is formed. You'll need 2 thumb tacks, string, paper and a pencil or a pen.

1. Secure the thumb tacks so that the distance between them is less than the length of the string.
2. Tie the string around both tacks
3. Pull the string taut with the pencil
4. Keeping the string taut, mark out the shape you get as you move the pencil around.

You should end up with an ellipse! And this is mathematically how we derive the standard form of the equation of the ellipse.

We simply require the distance formula for PF (the distance from point P to the positive focus) and PF' (the distance from the point P to the negative focus).

Thanks for watching. Please give me a "thumbs up" if you have found this video helpful.

Please ask me a maths question by commenting below and I will try to help you in future videos.

Integral of ∫sec^3(x)dx

Integrals of odd powers of sec(x) are never straight forward, unlike their even powered counterparts. They get exponentially more difficult as the odd power increases. The integral of sec³(x) is a classic integral that is still manageable by conventional analytical methods.

Probably the most straight forward approach is to use integration by parts. To do this, we can write the integral as:

∫sec³(x)dx = ∫sec(x)sec²(x)dx

So now we have the 2 parts required for I.B.P. if we let

u = sec(x) and dv = sec²(x)dx

Please watch the video to see how this integral is solved.

Suggested videos:

Integration of sec(x): https://youtu.be/9pG-1NG2Kqs
Derivative of sec(x): https://youtu.be/_BGccPnemDA

Integration of sec(x)

There is no obvious way to integrate the secant function, sec(x). One way is to use the method of partial fractions. While this method takes a quite a few steps to solve the integral, it is robust. It does not rely heavily in the intuition of the student.

Another way involves some trickery where we try to form an integral of the form f'(x)/f(x), which gives the result ln|f(x)|. Although this method is efficient, it requires you to know part of the answer before you begin, which doesn't make much sense.

Using the method of partial fractions, the idea is to rewrite the integral of sec(x) as:

∫sec(x)dx = ∫cos(x)dx / (1 - sin²(x))

Then by letting u = sin(x)...

∫sec(x)dx = ∫du / (1 - u²) = ∫du / (1+u)(1-u)

We can then convert the expression 1/(1+u)(1-u) into the sum of its partial fractions.

Suggested video:

Partial Fraction Decomposition

Conic Sections - Focus, Directrix and Eccentricity

You can construct a conic section on any plane by defining a fixed point called the focus (F), a moving point (P) and a straight line called a directrix D!

The locus of P will describe a conic section as long as the ratio of the distance PF (from the point P to the focus F) and the distance PD (perpendicular distance from the point P to the directrix D) is a constant, called the eccentricity (e = PF/PD).

The sections constructed from the locus depending on the value of e are:

1. Ellipse (0 < e < 1)
2. Parabola (e = 1)
3. Hyperbola (e > 1)

In this video, I show you an exercise that you can do at home to help you construct a conic section with conic graph paper!

To understand more about locus, please take a look at this page: Locus in the Cartesian Plane - Part 1