find a Cartesian equation for the conic section satisfying the given conditions. Sketch the curv

Find a Cartesian equation for the conic section satisfying...

Key Concepts

Ellipse

An ellipse is a conic section defined as the set of all points in a plane where the sum of the distances to two fixed points (the foci) is constant. This property distinguishes ellipses from other conic sections and is fundamental in deriving their Cartesian equations.

Vertices and Center of an Ellipse

The vertices of an ellipse are the points where the ellipse intersects its major axis, and the center of the ellipse is the midpoint between the vertices. The center serves as the origin for the ellipse’s standard equation and plays a crucial role in determining the orientation and dimensions of the ellipse.

Foci and the Relationship Among Parameters

In an ellipse, the foci are located along the major axis symmetrically about the center. The distances from the center to a vertex (denoted by 'a') and to a focus (denoted by 'c') are interrelated with the distance 'b' (the semi-minor axis length) by the equation c² = a² - b². This relationship is essential for finding the complete Cartesian equation of an ellipse.

Cartesian Equation of an Ellipse

Once the center and the lengths of the semi-major (a) and semi-minor (b) axes are determined, the ellipse can be expressed in its standard Cartesian form. For an ellipse centered at (h, k) with a horizontal major axis, the equation is ((x - h)² / a²) + ((y - k)² / b²) = 1. This formulation allows the ellipse to be analyzed and sketched accurately.

Transcript

00:03 And this problem, we're given the fosy and the vertices of an ellipse.

00:07 We ask to find the equation of the ellipse.

00:10 The first thing we want to recognize is that the second coordinate of the fosy and the vertices are the same values.

00:23 As a result, we know that our ellipse will be horizontal.

00:30 So if we have a horizontal ellipse, we know the form of the equation.

00:38 The form of the equation will be x minus x -subs -0 quantity squared divided by a squared plus y -minus -0 quantity squared divided by p squared equals 1.

01:00 So since we have a horizontal ellipse, we know the faux -side take on the following form.

01:06 X -0 minus c, comma, y -0, and x -0 plus c, comma, y -0, and x -0 plus c, comma, y -0.

01:18 By comparing coordinates, we can see that x of 0 minus c equals 3 and x of 0 plus c equals 5.

01:35 This will be a system of equations that we can now solve, recognizing that we have opposite coefficients on the c.

01:42 So if we add the two equations together, we will eliminate the c.

01:48 We get 2 times x0 equals 8.

01:54 Divide in both sides by 2, we get x -subs -0 equals 4.

02:00 Back substituting into either equation, we can solve for c.

02:06 In this case, i'm going to use the second equation.

02:11 So x -subs -0 was 4 plus c equals 5, so therefore c equals 1.

02:19 Though c is not in our equation of the ellipse, we will use the c to find one of the values in our ellipse equation in just a moment.

02:33 Next, if we compare the second coordinate for the fosi, we can see that y -subs -0 is equal to 2.

02:58 Next, we want to look at the form of the vertices for horizontal ellipse.

03:03 The vertices take on the following form, x -0 minus a, y, 0, and x -0 plus a, comma, y, zero.

03:15 By quick observation, if we compare the second coordinate, we can we can see that we still obtain that y sub 0 equals 2, same value we obtained by looking at the fosci.

03:28 Additionally, we can now compare the first components to see that x minus a equals 2 and x plus a equals 6.

03:40 In this case, we could again solve a system of equations as we did using the fosi.

03:46 But since we know the x of 0 value, we can use just one of the these values.

03:53 So i'm going to take the first one, x of 0 minus a equals 2.

04:02 Next, let's substitute our x of 0 value to be 4.

04:08 And next we can solve for a.

04:12 So let's subtract 4 on both sides of our equation.

04:15 So we have negative a equals 2 minus 4, which is negative 2.

04:19 Divide in both sides by negative 1, get a equals positive 2...