Jomo asks for a definition

A "parabola" is the set of all points that are equidistant between a give point and a given line. In partiular, if the line is the x-axis and the point is (0, a) then the distance from the point (x, y) to the line is y hile the distance to the point is \(\displaystyle \sqrt{x^2+(y- a)^2}\).

So, we must have \(\displaystyle y= \sqrt{x^2+ (y-a)^2}\). Squaring both sides, \(\displaystyle y^2= x^2+ (y-a)^2= x^2+ y^2- 2ay+ a^2\).

We can cancel the two \(\displaystyle y^2\) terms to have \(\displaystyle x^2- 2ay+ a^2= 0\). \(\displaystyle 2ay= x^2+ a^2\), \(\displaystyle y= \frac{1}{2a}x^2+ \frac{a}{2}\).
 
Jomo said:
Can someone please give me the definition of a parabola?
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Have you heard of a web-site called "Google"? (Their CEO is a super-smart kid from my Alma Mater!!)

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
Here is a picky note. But distance is a non-negative number. If the point is \((-6,2)\) its distance is \(6\) units from the \(y\)-axis.
In general the line need not be only vertical or horizonal. Given the line \(\ell: ~ax+by+c=0\) and point \(P: (a_0,b_0)\)
the distance from \(P\) to \(\ell\) is \(\dfrac{\left|ax_0+by_0+c\right|}{\sqrt{a^2+b^2}} \)
 
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