Find the vertices and the foci of the ellipse with the given equation. Then draw the graph. 5 x^

Find the vertices and the foci of the ellipse with the given...

Key Concepts

Ellipse

An ellipse is a type of conic section defined as the set of all points such that the sum of the distances from two fixed points (the foci) is constant. It is a closed curve characterized by its symmetry about two perpendicular axes.

Standard Form of an Ellipse

Converting the equation of an ellipse into its standard form, (x²/a²) + (y²/b²) = 1, allows for the direct identification of its key parameters. This form reveals the lengths of the semi-major and semi-minor axes, which are essential for understanding its geometry.

Vertices

Vertices are the points on the ellipse that lie at the endpoints of the major and minor axes. They represent the maximum extents of the ellipse along these axes and are found using the parameters a and b from the standard form.

Foci

The foci are two specific points located along the major axis of the ellipse. Their distance from the center is determined using the relationship c² = a² - b² (assuming a is the semi-major axis). These points are crucial in defining the ellipse's shape and its geometric properties.

Graphing Conic Sections

Graphing an ellipse involves plotting its center, vertices, and foci, then sketching a smooth closed curve that passes through the vertices. This process illustrates the symmetry and distinct geometry of the ellipse, and helps in visualizing how changes in the parameters affect its shape.

Transcript

00:01 5 x square plus 7 y square equals to 35 we divide both side by 35 and we get here that is x square by 7 plus y square by 5 equals to 1 and this can be written as here x square by root 7 whole square plus y square by root 5 equals to 1 here in a standard form of ellipse here that is x square by a square plus y a square by b square equals to one this is the standard form of equation of ellipse so here a equals to root 7 and b equals to root 5 here a is greater than b so here major axis is here horizontal neutral axis horizontal so x intercept x intercept that is here that is y coordinate 0 so minus root 7 0 and root 70 here so and y intercept here that is 0 minus root 5 and we also know that that is c s square equals to a is here is greater so c s square equals to a is greater so c is equal to a square minus b square and here that is a square minus b squared that is here root 7 whole square that is 7 minus 5 that is c equals to here 2 so c equals to so their fokey is here that is minus 2 0 and here because their point is major axis is horizontal so there are pokey here, this here.

02:03 And we have to plot the graph.

02:05 So this is x -axis.

02:06 This is negative x -tax.

02:08 And this is y, and this is negative y -d -s.

02:12 So this is taken to be zero.

02:14 And we have to take the point here.

02:17 So minus root 7.

02:19 That is root 7.

02:21 That is nearly to get a 2 point something here.

02:26 That is so.

02:28 So this is 1.

02:30 This is 2 and this is 3...