Derivation of the Equation of the Parabola

A circle is a locus of points  whose distance from a fixed point is a constant. A parabola can also be described as a locus of points whose distance from a fixed point and a fixed line not passing through that point is a constant. An example of a parabola is shown below.

In the figure below, point F is called the focus of the parabola and line l is called its directrix. The vertex of the parabola is at the origin O and the x-axis the perpendicular bisector of FH. If we take any point P(x,y) on the parabola, draw FP, and draw PQ perpendicular to line l where Q is line l, then the distance between F and P and P and Q are equal.

parabola

 

Suppose that the coordinates of the focus is (0,p) where p > 0, then the directrix is y = -p (can you see why?).

From here we can see that PQ = |y + p| and PF = \sqrt{x^2 + (y - p)^2 }. Since PQ is equal to PF,

|y + p| = \sqrt{x^2 + (y - p)^2 }

Squaring both sides, we have

(y + p)^2 = x^2 + (y-p)^2
y^2 + 2py + p^2 = x^2 + y^2 - 2py + y^2
2py = x^2 - 2py
4py = x^2
y = \frac{1}{4p}x^2.

This is the equation of the parabola with focus (0,p) and directrix y = -p.

 

Logarithm Base Changing Formula Proof

In a logarithmic expression, it is possible to change base using algebraic manipulation.  For example, we can change

log_416 to \dfrac{\log_216}{\log_24}.

In this post, we are going to prove why it is possible to do such algebraic manipulation. The change of base above can be generalized as

\log_ab = \dfrac{\log_cb}{\log_ca}.

Theorem

\log_ab = \dfrac{log_cb}{\log_ca}.

Proof

If we let \log_ab = x, then by definition, a^x = b.

Now, take the logarithm to the base c of both sides. That is

log_c a^x = \log_cb.

Simplifying the exponent, we have

x \log_ca = \log_cb.

Now, since a \neq 1, \log_ca \neq 0.

Therefore,

x = \dfrac{\log_cb}{\log_ca}

Thus,

\log_ab = \dfrac{log_cb}{\log_ca}

 

Year 2015 in Review – Complete List of Posts

It’s the end of the year again, so let’s look at what we have learned so far in 2015. Below is the complete list of posts of mathematical proofs this year. Enjoy learning!

Year 2015 in Review – Complete List of Posts

You may also want to visit the Post List Page to explore the complete list of posts of Proofs from the Book.