Shifting Conic Sections

Find the center, foci...

Question

Shifting Conic Sections

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.

9x² + 6y² + 36y = 0

Accepted Answer

Find the center, vertices, fossa, and axes links of the given ellipse. By 16 X2 + 25 Y2 + 200 Y equals 0. Now, to solve this, we need to rewrite this in standard form. So let's modify it a little bit. 16 X2 + 25 multiplied by Y2 + 8Y is what we can do first. This equals 0. Now we have to complete the square. To complete the square, we'll do this for the Y 2. We'll take the 8 on the Y, divide it by 2, and square everything. This gives us a value of 16. Meaning we have 16 x2 + 25 multiplied by y2 + 8Y + 16. Equals Now, on the right side of the equation, since we did it to the left, We're going to add from 0 a 25 multiplied by 16. Because we added 16 on the left, we also added to the right. And that 25 will also carry over. We now have 16 x2 + 25 multiplied by y2 + 8y + 16. Equals 400. From here we can simplify by factoring. Our trinomial y squared. 16 x 2 + 25 multiplied. By y + 42 equals 400. Finally, divide everything by 400 to get our standard form x 2. Divided by 25 plus Y + +42. Divided by 16. Equals one. Now, this is our standard form of the ellipse. And from here we can find our sinner. We know the sin our sinner will just be zero for the x. And -4 for the Y. We can also find our semi-major axis A, since this is horizontally inclined. A 2 equals 25, which means A equals 5. B2 equals 16, which means B equals 4. We can use this then to find C, which we'll use to find the first sign. C is the square root of A2 minus B2, so we have the square root of 25 minus 16. Which gives us a square root of 9 or 3. Now we can find all of our required values. Our center, once again, is 0-4. Our vertices, because this is a horizontal ellipse, are given by H+ or minus AK. Which gives us 0 plus or minus 5-4. Which simplify 25-4 and -5-4. Our foci are given by H plus or minus CK because this is horizontal still. 0 plus or minus 3-4 gives us 3-4. And -3 -4. Finally, Our major axis length. Is given by 28, which will just be. 10 And a minor axis length. Is given by 2B, which is just 8. We finally have everything for our answer, and this is the solution to our problem. OK, I hope to help you solve the problem, thank you for watching. Goodbye.

Shifting Conic SectionsFind the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.9x² + 6y² + 36y = 0

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Shifting Conic Sections

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections in Exercises 57-68.

9x² + 6y² + 36y = 0