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

Standard Form of an Ellipse Equation

The ellipse is typically written in standard form as (x - h)²/a² + (y - k)²/b² = 1, where (h, k) is the center of the ellipse and a and b are the semi-axis lengths along the x and y directions, respectively. This form makes it easier to identify the key parameters and geometrical features of the ellipse.

Center of an Ellipse

The center of an ellipse is the midpoint between its vertices and is represented by (h, k) in the standard form equation. It serves as the point of symmetry for the ellipse and is the basis for determining other key features such as the vertices and foci.

Vertices of an Ellipse

Vertices are the points where the ellipse intersects its major axis. They are found by moving a distance equal to the semi-axis length along the direction of the major axis from the center. The vertices help in defining the extent of the ellipse in the direction where it is longest.

Foci of an Ellipse

The foci are two fixed points located along the major axis of the ellipse such that for every point on the ellipse, the sum of its distances to the foci is constant. The distance from the center to each focus (c) is found using the relationship c² = a² - b², emphasizing the elliptical property of constant total distance.

Major and Minor Axes

The major axis of an ellipse is the longest diameter passing through its center and vertices, with a total length of 2a. The minor axis is the shortest diameter, perpendicular to the major axis at the center, with a total length of 2b. These axes are essential for understanding the shape and orientation of the ellipse.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, drawing the major and minor axes, marking the vertices and foci, and then sketching a smooth curve that reflects the symmetry about both axes. This visual representation aids in comprehending the spatial relationships and key features of the ellipse.

Transcript

00:01 We have x square plus y plus 2 whole square divide by 4 equals to 1 so in a standard form we can write this expression as x minus 0 whole square divide by 1 square plus y minus of minus 2 its whole square divided by 2 square equals to 1 as denominator of y is greater so what we can see it is an vertical ellipse as denominator of y square is greater so two greater than one so hence it is vertical ellipse now if we're comparing comparing with x square divided by one square plus y square divided by two square equals to one so here what is the center so this is sifted by given ellipse is shifted by 0 minus 2 units.

01:24 That means 0 units in x direction minus 2 units in y direction.

01:29 So the first part of this question we have to find center.

01:35 Center is 0 minus 2.

01:40 Vertices we have that is it is vertical lift so 0 plus 0 2 minus 2 and 0 plus 0 and 0 plus 0.

01:57 0 minus 2 and next is minus 2 which is 0 0 and 0 minus 4.

02:09 4 key we have before 4 key we should find the value of c so value of c is a square root of 4 minus 1 which is a square root of 3 so 4 key was 0 plus minus root 3 so it is 0 plus 0 root 3.

02:30 Minus 2 next is 0 plus 0 minus of root 3 minus of 2 which gives 0 root 3 minus 2 and 0 minus of root 3 minus of 2 length of major axis this is part b of this question major axis is 2 times a which is 2 times 2 4 units, length of minor axis is equal to 2 times b and b value is here 1 so 2 times 1 that is 2 units if we'll draw the hyperbola it is in case of ellipse in case of hyperbola we should know one thing one is the transverse axis and one is the conjugate axis but in the ellipse the 2 axis is called major axis and minor axis but it is a vertical lips here so, it is the case of major axis is vertical axis, minor axis would be the horizontal axis.

03:52 So let's say we have centered 0, let's say we are saying it is y xxas and here it is xxas.

03:59 Center is 0 minus 2...

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