In Exercises 37–50, graph each ellipse and give the location of i...

Question

In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

Equation of an ellipse in standard form: (x + 1)² + 4(y + 4)² = 100.

Accepted Answer

The given ellipse and indicate his X plus one squared plus four, multiplied by Y plus four squared is equals to 100. Now, to solve this, we have to have the standard form of an ellipse, X -8 Squared divided by a squared plus Y minus K squared divided by B squared as equals to one. Since we need to be equal to one, let's divide everything here by 100. We should then have X plus one squared Divided by 100 Plus four, multiplied by AY plus four squared Divided by 100 is equals to one which further simplifies. So X plus one squared divided by 100 plus Y plus four squared divided by 25 As equals to one. Now we notice we have our form with A squared is equals to 100 B squared is equal to 25. We can then determine this is a horizontally oriented ellipse. Since our A squared is bigger Then RB two. No, we can go ahead and find the end points of our lips to find our endpoints. Since we're horizontally aligned, we take H plus A OK. N H minus A Yeah, where H K is our center of our lips. Based on our equation, our center will occur at negative one negative four. We can then go over to find our end points as negative one Plus R.A. also be 100. I'm sorry a is 10, not 100. Our a is 10 Native one Plus 10. I do four, Neither 1 -10, negative four together end points of nine, And -11 -4. Now we have our end points, we can find our focus or a sign since we have two of them, no side. Since this is horizontally aligned, you will take the equations H plus C K N H minus C K or C what will be the square root A squared minus B squared which is the linear eccentricity of our ellipse. We know a squares 100 B squares 25. We should get C is equals to the square root of or five squared of three. Using our two equations. You will get negative one plus five squares of three comma negative the four and A one minus five squares of three, negative four. We're in to simplify, We will get negative one Plus five squids of three which equals 7. a four -1 -5 vs three, which gives us negative 9.66 negative four. We have our first time and we have our end points looking at our possible graphs we can then determine the answer for our problem is answer B just all the points we found. OK. I hope to help you solve the problem. Thank you for watching. Goodbye.

In Exercises 37–50, graph each ellipse and give the location of its foci. (x +3)²+ 4(y -2)² = 16

Key Concepts

Standard Form of an Ellipse

Foci of an Ellipse

Graphing an Ellipse

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