Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola. Then sket

step 1 So the key to this problem is being able to complete the square. And what you'll also notice is

Find the center, vertices, foci, and the equations of the...

Key Concepts

Hyperbola

A hyperbola is a type of conic section that appears as two separate curves called branches. It is defined as the set of all points in the plane where the absolute difference of the distances to two fixed points (foci) is constant. The hyperbola's orientation, shape, and position depend on the coefficients and constant terms in its equation.

Center of a Hyperbola

The center of a hyperbola is the midpoint of the line segment joining the vertices. It acts as the point of symmetry for the hyperbola and is also the intersection point of the asymptotes. Computing the center is essential as it facilitates the translation of the hyperbola to its standard form.

Vertices

The vertices of a hyperbola are the points located on the transverse axis where the hyperbola intersects this axis. They represent the closest points between the two branches of the hyperbola and are found by determining the distance from the center along the axis aligned with the opening of the hyperbola.

Foci

The foci are two specific points located along the transverse axis of a hyperbola. Unlike the center or vertices, the foci do not lie on the hyperbola itself; however, they are crucial for defining the hyperbola because the difference in distances from any point on the hyperbola to the foci remains constant.

Asymptotes

The asymptotes are straight lines that the hyperbola approaches as it extends to infinity. They pass through the center of the hyperbola and provide a framework or guide for sketching the curves. The slopes of the asymptotes are determined by the relationship between the distances along the transverse and conjugate axes in the hyperbola’s standard equation.

Completing the Square

Completing the square is an algebraic technique used to rewrite quadratic equations in a form that reveals the geometric properties of conic sections, such as the center, vertices, and orientation. This method is particularly useful when transforming the general equation of a conic into its standard form, thereby making it easier to identify and analyze the conic's features.

Transcript

00:01 So the key to this problem is being able to complete the square.

00:04 And what you'll also notice is that x squared is by itself.

00:11 That's y squared plus 36y minus 72 is equal to zero.

00:21 So what i would do is because every teacher is a little different is i would first add 72 to the right side and i would factor out of negative nine from both the y squared and the 36 and you can double check that this is equal if i were to distribute that negative 9 in there would equal what's right there and then i leave a space right here because i'm going to complete the square which is taking half of this number negative 4 squared is equal to positive 4 so that's what i'm going to add right here but i'm not really adding 4 to the problem i am subtracting negative 9 times, well, i guess i say adding negative 9 times 4 would be 36.

01:09 So if i remove 36 from the left side, i also have to remove 36 from the right side.

01:15 So as i'm looking at that problem, 702 minus 36 is 36.

01:20 And what i have on the left side is 9 times a perfect square trinomial.

01:25 So it's y minus 2 squared.

01:28 I'm almost there.

01:30 What i need to do is get this equation equal to 36, which i can do by dividing every piece by 36.

01:38 And it might even help you to rewrite this as x minus zero squared over six squared.

01:44 And then nine goes into 36 four times, but i'm going to write it as two squared.

01:49 And you can double check that this math is correct...

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