Finding an Equation of a Hyperbola Find an equation of the hyperbola such that for any point on

Finding an Equation of a Hyperbola Find an equation of the...

Key Concepts

Standard Form of a Hyperbola

The standard equation of a hyperbola is typically presented in forms that depend on its orientation. For a hyperbola opening horizontally, the equation is of the form (x ? h)²/a² ? (y ? k)²/b² = 1, where (h, k) is the center, a is the distance from the center to a vertex along the transverse axis, and b is related to the distance from the center to the asymptotes. Understanding this form is crucial for translating the geometric definition into an algebraic equation.

Transverse Axis

The transverse axis of a hyperbola is the line segment that passes through the center and both foci. It determines the direction in which the hyperbola opens. The constant difference in distances between any point on the hyperbola and the foci is directly related to the length of the transverse axis.

Foci

The foci are two fixed points used in the definition of a hyperbola (and an ellipse). For a hyperbola, each point on the curve has distances to the foci that differ by a constant amount. The positions of the foci determine the orientation and shape of the hyperbola, as well as its other geometric properties.

Hyperbola Definition

A hyperbola is a type of conic section defined as the set of all points in a plane such that the absolute difference of the distances from any point on the curve to two fixed points (called foci) is constant. This definition distinguishes hyperbolas from other conic sections like ellipses, which have a constant sum of distances.

Transcript

00:01 Okay, what we want to do is to find an equation of a hyperbola such that for any point on the hyperbola, the differences between the distance from two points is six.

00:15 And the two points that we're interested in is the points 10, comma, two, and 2, 2.

00:27 And also we are told, nope, and that's it.

00:36 And so since the y values are the same in the points, the transverse axis is the horizontal.

00:58 And these are the foci.

01:05 And it's because if you look at the explanation for hyperbola, the differences between the distance of the of the fosy is a constant value.

01:24 Okay.

01:25 And so those are the fosi.

01:29 And so if these are the fosi, then we also know that the fosi, then we also know that the fosi, then we also know that the fosi.

01:35 The center is exactly in the middle of those two, which makes the center six, two.

01:43 Okay.

01:44 And then also what we want to look at by definition that the distance, that distance between those two points that are actually the foci, is also twice the value of a will equal that distance as well.

02:15 And so this is in the calculus book, in the book, that says that twice the distance, and that's because the transverse axis is horizontal.

02:27 If the transverse axis was vertical, then it would be 2b is equal to that distance.

02:33 Okay...

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