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

Major and Minor Axes

The major and minor axes are the longest and shortest diameters of the ellipse, respectively. The length of the major axis is 2a and the length of the minor axis is 2b, where a is greater than or equal to b. Determining these lengths is essential for understanding the overall size and elongation of the ellipse.

Graphing an Ellipse

Graphing an ellipse involves plotting its center, vertices, and foci, and then sketching a smooth curve ensuring that the shape reflects its standard geometric properties. This process helps visualize the scale and orientation of the ellipse based on its algebraic equation.

Foci of an Ellipse

The foci are two fixed points along the major axis of the ellipse that collectively define its shape. They lie closer to the center than the vertices, calculated using the relationship c² = a² ? b², where c is the distance from the center to each focus.

Center of an Ellipse

The center of an ellipse is the midpoint between its vertices and indicates the point around which the ellipse is symmetrically distributed. In the standard equation, the center is given by the coordinates (h, k), which result from the horizontal and vertical translations of the ellipse.

Vertices of an Ellipse

Vertices are the points where the ellipse intersects its major axis, representing the farthest horizontal or vertical extent of the curve relative to the center. They are determined by moving a distance of ‘a’ (the semi-major axis length) from the center along the major axis.

Standard Form of an Ellipse

The standard form of an ellipse’s equation is expressed as ((x ? h)²)/a² + ((y ? k)²)/b² = 1 or its vertical counterpart, where (h, k) is the center of the ellipse and a and b are the semi-major and semi-minor axes respectively. This form makes it easier to identify key characteristics of the ellipse such as its center, orientation, and dimensions.

Transcript

00:01 We have x plus 5 whole square divided by 16 plus y minus 1 whole square divided by 4 equals to 1 we will compare this equation with x minus h whole square divided by a square plus y minus k whole square divided by b square equals to 1 what we are getting here the center is h comma k vertex we have that is a plus h comma 0 plus k and a minus h so minus a plus h zero plus k for key we have c plus h zero plus k and minus c plus h zero plus k.

01:10 So we should find these things of what if we compare these two equations what we are getting the value of a is 4 value of b is 2 value of c is square root of a square minus b square that is square root of 12 which is 2 root 3 so party of this question center is minus 5 1 next, vertices is it is a right so a is 4 minus 5 0 plus 1 and minus 4 minus 5 0 plus 1 which is minus 1 and minus 4 minus 5 0 plus 1 which is minus 1 1 and minus 9 1.

02:17 Key we have c plus h 0 plus h minus c plus s 0 plus h so 2 root 3 plus 1 so 2 root 3 minus 5 1 and minus of 2 root 3 minus of 5 1 part b length of major axis is 2 times a so a value is 4 so 2 times 4 is 8 units and length of minor axis is 2 times p.

03:09 So b value is 2, so 2 times 2 is 4 units.

03:15 If you draw this curve so it is y -axis and let's say this is we have to go up to minus 9.

03:25 So x -axis will take a little bit above.

03:43 This is x -axis.

03:45 The vertex we have, if we'll focus on the vertex, center minus 5, minus 1, so minus 1, minus 2, minus 3, minus 4, minus 5.

03:55 So x is minus 5, y is minus 1.

04:02 We should highlight on x direction.

04:07 We should take y -axis and x -axis.

04:17 This is x.

04:27 Center is minus 5 .1...