The area (in sq units) of the quadrilateral formed by the...

Key Concepts

Tangent Line

A tangent line to a conic section like an ellipse is a straight line that touches the curve at exactly one point without cutting through it. The properties of tangents, including their slopes and points of contact, are critical for solving problems involving intersections and determining geometric figures formed by multiple tangents.

Quadrilateral Area Calculation

Calculating the area of a quadrilateral formed by intersecting lines in a coordinate plane typically involves finding the coordinates of the vertices by solving the equations of the lines involved and then applying methods such as the shoelace formula. This process integrates fundamental coordinate geometry techniques with algebraic solutions.

Ellipse

An ellipse is a type of conic section characterized by the set of all points for which the sum of the distances to two fixed points (the foci) is constant. It can be represented by a standard equation in Cartesian coordinates, and its geometric properties, such as axes, foci, and eccentricity, are fundamental in understanding its shape and behavior in a plane.

Latus Rectum

The latus rectum of an ellipse is a chord that passes through one of the foci and is perpendicular to the major axis. Its length relates directly to the eccentricity and the distances from the center to the ellipse along the minor axis, playing a key role in understanding the focus-directrix properties of the ellipse.

Transcript

00:01 There are two parts to problem 35, and i'm just going to begin with the first part, which is simply finding the equation of the tangent line that is at 0 .24 on the ellipse x squared plus 3y squared equal to 52.

00:15 I'm going to use the equation x1x over a squared plus y1y over b squared equal to 1.

00:25 And what i can go ahead and do is put my ellipse in standard form, and then i can go ahead and plug in the values for x1 and y1.

00:33 I'm going to take my ellipse, put it in standard form, so i have x squared over 52, that's going to be plus 3y squared over 52, equal to 1.

00:44 Now i'm going to go ahead and i want to leave that 3 in the numerator, and you normally wouldn't do that, but that's just going to make life a little bit easier.

00:56 So it doesn't really matter in this specific application.

01:00 So i can go ahead and put in my values for x1 and y1.

01:04 So i have 2x over 52.

01:09 That's going to be plus 4.

01:12 And remember i have that 3 in the numerator.

01:14 I made sure to point that out.

01:16 That's going to be times y over 52 equal to 1.

01:20 Now i want to go ahead and 52 i can indeed, has a common factor with 2, which is 2.

01:28 That means that's going to simplify in the denominator to 26.

01:32 And similarly, on the right -hand side, that 4 is going to become 2, and that 52 is going to become 26 with the y, which means that both these have 26 in the denominator.

01:43 You multiply this whole equation by 26, which will mean that i have x is going to be plus 2 times 3 times y, which is going to be 6y, that's going to be equal to 26.

01:57 So that was just several steps of algebra in one line right there.

02:02 So you can break that apart and maybe work that through in separate steps, but you should end up with this result.

02:09 And i want to put in the form y is equal to mx plus b.

02:12 I have 6y equal to negative x plus 26.

02:17 And then i want to divide each side by six, so i get y all by itself.

02:23 So y is going to be equal to negative 1 -6.

02:27 Plus 26 over 6.

02:30 Well, 26 does indeed divide by 2.

02:33 That means it's going to be 13 over 3.

02:38 That is our, oops, negative 1, 6x.

02:41 I almost forgot the x...