An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b)

An equation of an ellipse is given. (a) Find the center,...

Key Concepts

Graphing an Ellipse

Graphing an ellipse involves plotting its center, vertices, and foci, and drawing a smooth curve that reflects the ellipse's symmetry about both axes. This step is important as it provides a visual representation, helping to interpret the algebraic results geometrically and understand the ellipse's spatial properties.

Major and Minor Axes

The major and minor axes of an ellipse are the longest and shortest diameters, respectively. The lengths of these axes are 2a and 2b, where a is associated with the major axis. These measures describe the overall size and elongation of the ellipse, and comparing their lengths reveals the ellipse's eccentricity.

Completing the Square

Completing the square is an algebraic method used to transform a quadratic equation into a perfect square trinomial form. This technique is essential when rewriting conic section equations from their general forms into standard forms so that the geometric properties, such as the center and radii, can be easily determined.

Standard Form of an Ellipse

The standard form of an ellipse is expressed as ((x-h)²/a²) + ((y-k)²/b²) = 1, where (h, k) represents the center of the ellipse, and a and b denote the semi-axes. This form simplifies the process of identifying important features and reading off parameters directly from the equation.

Center, Vertices, and Foci

The center of the ellipse is found directly from the standard form as the point (h, k). The vertices, which lie along the major axis, are determined using the distance 'a' from the center, and the foci are located by calculating a distance 'c' from the center using the relationship c² = a² - b². Identifying these points is critical in understanding the shape and spatial orientation of the ellipse.

Transcript

00:02 Now phillips we have 9 x square minus of 54 times x plus y square plus 2y plus 46 is 0.

00:18 So we'll try to convert in a standard form.

00:21 So what we are getting we can write 9 common x square minus 6x plus we can write the 6 2 times 3.

00:31 So 2 times 3 is 9 minus 9 plus y square plus 2 y square plus 2 y.

00:38 Plus 1 minus 1 plus 46 is 0.

00:44 Here we can see it is 9 times x minus 3 whole square plus y plus 1 whole square minus 81.

00:56 It is minus 1 plus 46 which is equals to 0.

01:04 This gives us 9 times x minus 3 whole square plus y plus 1 whole square equals to 81 plus 1 whole square 1 is 82, 82 minus 46, we have 36.

01:23 And a standard form of this ellipse we are getting x minus 3 whole square divided by 36 by 9 is 4 plus y plus 1 whole square divided by 36 equals to 1.

01:37 And we can see the denominator of y plus 1 whole square is 36 which is greater than denominator of x squared term.

01:43 So it is in vertical ellipse and it will compare with x minus h whole square divide by b square plus y minus k whole square divide by a square equals to one.

02:08 So what we are getting the value of a we have 6 b is 4 and the point c which is a square minus b square that is 36 minus 4.

02:22 Is 32 16 times 2 that is 4 root 2 party of this question center is h comma k which is 3 minus 1 vertex is it is vertical lips of vertex is 0 plus k h comma plus minus a plus k which gives the value 0 plus h that is always center is 0 plus h is 3 so 3 comma it is center is minus 1 and a value is 6 so plus minus 6 minus 1 so 1 is 6 plus 6 minus 1 is 5 and 3 comma it is minus 6 and k is minus 1 so minus 6 minus 1 is minus 7 4 key we have it is 0 plus minus c so 0 plus h comma plus minus c plus k which we are getting 0 plus s is always 3 comma c is 4 root 2 minus 1 and 3 minus of 4 root 2 minus of 1 part b of this question length of major axis is 2 times a which is 6 times 2 is 12 units length of minor axis which is 2b that means 2 times b is 4 so 2 times 4 is 8 units we'll reflect this equation on the curve so where the vertex is 353 minus 7 this is y xx and 1 2 3 4 5 6 7 so this is x axis it is origin so 3 comma 5 center is 3 minus 1 1 2 3 this is the center vertex 3 5 so it is 0 5 so it is 0 6 6 0, 1, 2, 3, 4, 5, this is the first vertex...

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