Graphing Conic Sections

Find the eccentriciti...

Question

Graphing Conic Sections

Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.

5y² − 4x² = 20

Accepted Answer

Find the eccentricity of the hyperbola, 8Y2 minus 4 X2 equals 32. Sketch the hyperbola showing the foci vertices and asymptotes in the diagram. Now, for this, we first need to convert this into standard form. So let's take our equation 8Y2 minus 4x2 equals 32 and divide everything by 32. We end up getting Y2 divided by 4 minus X2 divided by 8 equals 1. Now this is a vertical hyperbole because Y2 is first. We have A2 equals 4, and B2 equals 8, meaning A equals 2. And Is the spirit of 8s. Now, we can use this to find the vertices. So we know this is A equals 2, and because it's a vertical, we will add and subtract 2 from the Y coordinate. So we will say our vertices. R 02 and 0-2. Now we can find that C. This will help us find the eccentricity. C is given by C2 equals A2 + B2 in hyperbolus. This will be given by 4 + 8, which is 12. That means c is the square root of 12, or 2 squares of 3. We can now use this to find eccentricity e which is given. By c divided by a or 2 squares of 3. Divided by 2. Giving us a square root of 3. So we now have our eccentricity. From here, we need a graph. We can find our full sign. Which are given By 02 skirts of 3. And 0-2 23. And we can find our asymptotes. Are asymptotes then Are given by y equals plus or minus a divided by b multiplied by x. We get Y equals plus or minus 2 divided by the square root of 8. X Which simplifies 2 plus or minus 22 divided by 2 X. Now from here we can draw our graph. I'll use blue for this. And we know the sinner is at 00. We have one vertex at 02 and another at 0-2. One of our low sci is at 0223. And 0-2 square of 3. We can now graph our asymptotes. So we'll put a point at a point on our line, and we just draw a dashed line for our first asymptote. This one will be Y equals 22 divided by 2 X. We'll do the same for our other asymptote. Which goes With a positive slope This is Y equals 22 divided by 2 X. We have our two fossa and our vertices. We now just have to draw our curves for the hyperbola. Which we just follow our asymptotes and make. A curve OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Graphing Conic SectionsFind the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.5y² − 4x² = 20

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Graphing Conic Sections

Find the eccentricities of the ellipses and hyperbolas in Exercises 59–62. Sketch each conic section. Include the foci, vertices, and asymptotes (as appropriate) in your sketch.

5y² − 4x² = 20