Use the Distance Formula to show that the equation of the parabola with focus F(0,1) and directr

step 1 focus is at the point zero one. Our directrix is negative one, y equals negative one. And then w

Use the Distance Formula to show that the equation of the...

Key Concepts

Distance Formula

The distance formula is a mathematical expression used in coordinate geometry to determine the distance between two points in a plane. It is derived from the Pythagorean theorem and is fundamental in relating geometric figures to algebraic equations.

Parabola

A parabola is a conic section characterized by its unique geometric property: every point on the curve is equidistant from a fixed point called the focus and a fixed line called the directrix. This definition is central to understanding the shape and algebraic representation of a parabola.

Focus-Directrix Property

The focus-directrix property is a defining characteristic of conic sections, especially parabolas, where each point on the curve maintains equal distance from a given focus and a corresponding directrix. This property allows for the derivation of the parabola's equation through setting the distance from a variable point on the curve to the focus equal to its perpendicular distance to the directrix.

Transcript

00:03 Focus is at the point zero one.

00:05 Our directrix is negative one, y equals negative one.

00:09 And then we're given that equation of the parabola.

00:12 So if i just sketch this out really quickly, make these our ones, and then i'm gonna do one, two, three, just to break that into 1 1 4th.

00:26 So my focus is at zero one.

00:29 That'll be the point right here.

00:31 And my directrix is the line y equals negative one, this line right here.

00:37 Now, if i wanted to sketch out a really quick parabola, i'm just gonna make a table.

00:46 So if i plug in one for x, that would be 1 4th times one.

00:53 So y would be 1 4th.

00:55 I plug in zero, zero for x squared, i'd have 1 4th times zero, which is zero.

01:02 And then if i do negative one, that squared would be positive one, and i would get 1 4th again.

01:09 So very quickly, that means that my parabola has a vertex at the origin, it's going to be 1 1 4th, negative one, 1 4th.

01:20 That's why i made those little red lines there.

01:22 I was breaking apart so i could go by fourths.

01:29 So that's just a very rough sketch for my parabola.

01:31 But the reason why a parabola is so special is that the distance between the focus and a point on the parabola is the same distance from the parabola to the directrix.

01:41 So we can use the distance formula for this problem here.

01:47 And so our distance formula, the distance is equal to x2 minus x1 squared plus y2 minus y1 squared...

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