Proof for the definition of parabola??

Question

In my maths textbook parabola is defined as path traced by a point which moves in a plane in such a way that its distance from fixed point (focus) is always equal to its distance from a fixed line(directrix) both lying in the same plane where as the given fixed point does not lie on the given line.

Is there any proof for this definition or we accept it as an axiom ?

Answer 1

I'm going to guess that what you are actually asking is how this definition leads to the equation of a quadratic function that you're used to seeing for a parabola. For simplicity's sake, let's assume that the focus lies on the y-axis, say its coordinates are $\,(0,\alpha),\,$ and that the directrix is the horizontal line $\,y=-\alpha,\,$ with $\,\alpha \neq 0\,$ so that the focus is not on the directrix.

Let $\,P(x,y)\,$ be a point satisfying the given condition. Then the distance from $\,P\,$ to the focus is given by $\,\sqrt{x^2+(y-\alpha)^2};\,$ and since the directrix is horizontal, the distance to it from $\,P\,$ will be purely vertical, i.e. $\,\left|y-(-\alpha) \right| =\left|y+\alpha\right|=\sqrt{(y+\alpha)^2}.\,$ Then we have that

$$ \begin{align} \sqrt{(y+\alpha)^2} &= \sqrt{x^2+(y-\alpha)^2}\\ (y+\alpha)^2 &= x^2+(y-\alpha)^2\\ (y+\alpha)^2 - (y-\alpha)^2 &= x^2\\ 4\alpha y &= x^2\\ y &= \frac{1}{4\alpha}x^2\\ \end{align} $$

Letting $\,a=\frac{1}{4\alpha}\,$ gives the familiar quadratic equation $\,y=ax^2.\,$

Every other parabola can be thought of as an expansion/compression, rotation and/or translation of this simple parabola, so by applying the appropriate transformations to the focus and directrix we can generate the equation of every parabola in the same manner.

Answer 2

It's not an answer. Let me add some comments. First of all, please note that one of the simplest ways of defining conic is the following:

Let S be a fixed point on the plane, and AB be a straight line that does not pass the given fixed point S. Now, consider the locus of a point P such that SP/PM is constant, where MP is the perpendicular distance from P to the line AB. The locus of P is defined as a conic.

The fixed point S is called the focus, the line AB is called the directrix, and the ratio SP/PM is called the eccentricity of the conic. Many people denote SP/PM by e.

Note that if e= 1, then we say, our conic is a parabola. Similarly, we can define hyperbola and ellipse.

By the way, the phrase "a path traced by a point" is replaced the fancy word "locus".

I hope this helps.