Find an equation of the curve on which point P lies. The difference of the distances between P(x

Find an equation of the curve on which point P lies. The...

Key Concepts

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in the plane. It is instrumental in formulating the condition of the constant difference in distances that defines a hyperbola.

Hyperbola

A hyperbola is a type of conic section defined as the set of all points in the plane such that the absolute difference of their distances from two fixed points (called foci) is constant. This characteristic directly leads to its canonical equation and geometric properties.

Locus Definition

In geometry, a locus is the collection of points that satisfy a particular condition or set of conditions. In this problem, it refers to the set of all points P(x, y) that maintain a fixed difference in distances from two designated points.

Transcript

00:01 Well, we're ready to write another question that's going to deal with the definition of a hyperbola.

00:07 And remember the definition for a hyperbola.

00:10 We have two fixed points.

00:11 I'm going to tell you the two fosci.

00:13 One is at 3 -91, and the other one is at 3 -5.

00:21 And remember for a hyperbola that our definition, sorry, i'm not very good at drawn straight lines here, that if i go to my fixed point 3 -9 -1, again, i'll say that's the first focus, and then 1, 2, 3, 4, 5, go up here to the second focal point and have that at 3 -5.

00:52 We, for our definition, we know that, for one thing, this hyperbola is going to end up graphing upward and downward.

01:05 Exactly where the vertices are we don't know yet, but we will know those very shortly because i'll give you a little bit more information.

01:12 Remember the definition of a hyperbola says that if you take a point on the hyperbola and you calculate the distance from that point to a focus and the distance of the other point to that focus, that the difference between those two is a constant.

01:32 So the difference between the distance from the point on the hyperbola to each fosci subtracted that that difference is equal to a constant.

01:43 And let's say i tell you that that difference is equal to five.

01:47 And remember, that difference is actually equal to 2a.

01:52 Therefore, we actually know that a is five halves or two and a half.

01:57 And we know where the center is.

01:59 We know that the distance between these two points is six units.

02:05 And so if we drop down half that much, one, two, three, right there at the point three and two, that is going to end up being the center for our hyperbola.

02:18 And we know that this is the distance we move to get to a vertices.

02:22 So actually, if we move up two and a half units and down two and a half units, those are the location of my vertices.

02:30 And i can see that if i'm going to write the equation for this hyperbola, i'm going to need to put my y term as the lead coefficient.

02:37 So i'm going to have y minus 2 quantity squared over...

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