Show that the ellipse x^2+2 y^2=2 and the hyperbola 2 x^2-2 y^2=1 intersect at right angles.

Name: Problem 27, Chapter 2: Calculus: A Complete Course
Uploaded: 2021-11-17T13:18:56-08:00
Duration: 3 min 27 s
Channel: Gregory Higby
Description: Show that the ellipse x^2+2 y^2=2 and the hyperbola 2 x^2-2 y^2=1 intersect at right angles.

Show that the ellipse x^2+2 y^2=2 and the hyperbola 2 x^2-2...

Key Concepts

Implicit Differentiation

Implicit differentiation is a process of finding the derivative of a function when it is not given in the explicit y = f(x) form but rather as an equation involving both x and y. By differentiating both sides of the equation with respect to x and applying the chain rule to terms involving y, one can solve for dy/dx, which represents the slope of the tangent line at any point on the curve.

Tangent Lines and Orthogonality

The tangent line to a curve at a specific point conveys the instantaneous direction of the curve. Two curves are said to intersect orthogonally if their tangent lines at the point of intersection are perpendicular, which mathematically means that the product of their slopes (dy/dx) is -1. This concept is fundamental in understanding how curves meet and interact.

Intersection of Curves

The intersection of curves occurs when the points (x, y) satisfy the equations of both curves simultaneously. Determining these points is essential for analyzing the relationship between the curves, such as verifying whether they intersect at right angles.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone and include ellipses, hyperbolas, parabolas, and circles. Each of these curves is represented by a quadratic equation in two variables, and their geometric properties, such as symmetry and curvature, are essential when studying intersections and angles between them.

Transcript

0:00 All right.

00:01 There's actually a lot of steps to go into this problem, but it's a beautiful problem.

00:08 So we have this piece, and then we also have 2x squared minus 2y squared is equal to 1.

00:18 And the first thing i need to do is find the points of intersection.

00:21 Now i'm going to use elimination because these two things are just canceled.

00:24 So i'm adding straight down, adding straight down, and then when you square root, or so you got divide by 3 first.

00:31 Then when you square root, you get the points of intersection, i'm going to be at positive and negative.

00:35 One.

00:38 But that's not enough.

00:39 You also have to figure out what the y coordinates would be.

00:41 So if i said, you know, x equal to 1, you know, x squared equals 1 right here and plug it back in, you have 1 plus 2, y squared is equal to 2.

00:51 So if you subtract the 1 over and then divide by 2, and whether or not you rationalize this is up to you, but if you rationalize the denominator, it's root 2 over 2.

01:05 So now what you have to do is find the slopes of each of those.

01:08 So the slope of the ellipse would be 2x plus 4y, d, y, d, y, d, x is equal to zero, the derivative to zero.

01:20 So what i have to do is plug in, i forgot right plus or minus up here, but i'm only going to do a few of these, like when x is one, so two times one, and then when y is root 2 over 2, you know, 4 divided by 2, 2 is 2, so it's 2 root 2, d, y, d, d, x...

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