Sketching an Ellipse In Exercises 25-30 , find the center, foci, vertices, and eccentricity of t

Sketching an Ellipse In Exercises 25-30 , find the center,...

Key Concepts

Ellipse

An ellipse is a type of conic section defined by the set of points for which the sum of the distances from two fixed points (the foci) remains constant. It is characterized by a smooth, closed curve that exhibits symmetry about both its major and minor axes.

Standard Form of an Ellipse

The standard form of an ellipse’s equation is essential for analyzing its properties easily. It is typically written as (x - h)²/a² + (y - k)²/b² = 1, where (h, k) represents the center of the ellipse, and a and b represent the lengths of the semi-major and semi-minor axes respectively, providing a clear framework to locate key geometric features.

Completing the Square

Completing the square is an algebraic method used to transform quadratic expressions into perfect square trinomials. This technique is crucial for rewriting the general quadratic equation of a conic section into its standard form, thereby simplifying the process of identifying the center, axes, vertices, and foci.

Center, Vertices, and Foci

The center of the ellipse is the point that serves as a reference for other features, acting as the midpoint between the vertices on the major axis. The vertices mark the furthest points along the ellipse's principal axis, while the foci, located along the same axis, are key in defining the ellipse through the constant sum of distances property.

Eccentricity

Eccentricity measures the degree of deviation of an ellipse from a perfect circle. It is calculated as the ratio of the distance from the center to a focus and the distance from the center to a vertex. Lower eccentricity values indicate an ellipse that is nearly circular, whereas higher values denote a more elongated shape, thus providing insight into the shape and behavior of the ellipse.

Transcript

00:01 To find the center, bauxite, vertices, and eccentricity of this ellipse, and ultimately graph it, it's going to be helpful to rewrite it in standard form first.

00:12 So just to begin by writing out the equation of the ellipse again.

00:16 We have x squared plus 10 y squared minus 6x plus 20y plus 18 equals 0.

00:32 And to get this into standard form, we're going to need to complete the square.

00:37 So to complete the square, let's group the x and y term together.

00:42 So x squared plus 6x, and for the y terms we're also going to need to factor out that 10.

00:50 So factor out of 10, and factoring a 10 out of this term leaves the y squared, factoring out of 10 over the other y term, use the 2y, and then plus 18.

01:06 Equals 0.

01:08 So now we're ready to complete the square for the x terms.

01:13 We're going to need an add a 9.

01:15 Half of 6 is 3, 3 squared 3 squared is 9, so x squared plus 6 plus 9, plus 10.

01:27 And i'm completing the square inside this factor here.

01:30 We have y squared plus 2y, and the number we need to add is 1, plus 18, equals, and now, because we've added a 9 to this, the equation, we'll add a 9 to this side, and here we added a 1, but really we're actually adding a 10, because we have 1 times 10 if we flew to distribute, so we've added a 10.

01:57 So adding 9 here, adding 10 here, add 9 and 10 to this as well.

02:02 So now let's factor this, so we'll have x plus 3 squared plus 10 times y plus 1 squared.

02:19 I just noticed a typo, my mistake.

02:21 This should be a minus 6x, not a plus 6x.

02:25 I just want to change some of these plus signs to a minus sign.

02:31 It's going to be minus signs.

02:33 It should be a minus here.

02:38 The minus, minus, minus 3.

02:40 Okay.

02:41 And then over here equals, let's move the 18 over and combine these two.

02:45 So that's just going to make a 1.

02:47 Now, this is almost in standard form.

02:51 Standard form for an ellipse, usually there's a number underneath the x and y terms.

02:56 So x minus 3 squared what would you put in the denominator we're just going to put a 1 there just sort of a placeholder to keep track of our constants and how can we reinterpret there's 10 multiplying by 10 is the same as dividing by 1 tenth so we're going to put y plus 1 squared divide by 1 tenth and still equal to 1 okay now that we have this form we can go ahead and state with the center of the ellipses this is x minus h, this is y minus k, and the center is at h comma k, so the center is at 3, negative 1, the center of the ellipse.

03:39 Next up, we need to find the vertices of the limits.

03:44 To do that, let's state what a and b are.

03:46 Well, we really only need a, but we're going to state b also.

03:50 So this number here is a squared.

03:53 A squared equals 1.

03:55 This means a equals 1.

03:57 On a and b are usually positive values.

04:00 Just for the sake of doing the algebra.

04:04 And b squared is 1 tenth.

04:09 We're going to use this later, not right now, but for completion.

04:13 We're going to talk for b.

04:15 When we square each side and rationalize, we'll get square of 10 over 2.