Consider the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 where half the length of the major axis is a , an

step 1 So we're given the following information about an ellipse. This ellipse is described by the equa

Consider the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 where half...

Key Concepts

Distance Ratios in Conics

Distance ratios in conic sections refer to the constant proportion that exists between the distance from a point on the conic to a focus and its distance to the directrix. This fundamental property is used to derive and prove various geometric relationships within the ellipse, such as the connection between the focal distance, the directrix, and the eccentricity.

Directrix

A directrix is a fixed line associated with a conic section. It serves as one of the defining elements in the alternative definition of conics where each point on the conic maintains a constant ratio of its distance to the focus and its distance to the directrix. In the context of ellipses, identifying a directrix helps in establishing relationships between distances and the eccentricity.

Eccentricity

Eccentricity is a parameter that quantifies how much a conic section deviates from being circular. For an ellipse, the eccentricity (denoted by e) is defined as the ratio of the distance from the center to a focus (c) to the semi-major axis (a), with the property that 0 < e < 1. It plays a vital role in relating the distances from any point on the ellipse to the focus and to the directrix.

Ellipse

An ellipse is a type of conic section defined as the set of all points in a plane such that the sum of the distances to two fixed points (called the foci) is constant. The standard form of the equation, x²/a² + y²/b² = 1, highlights the relation between the semi-major axis (a), the semi-minor axis (b), and the geometric structure of the curve.

Foci

The foci are two fixed points located along the major axis inside the ellipse. Their positions are essential in the geometric definition of the ellipse since every point on the ellipse maintains a constant sum of distances to these points, making them critical in understanding the ellipse’s properties.

Transcript

00:01 So we're given the following information about an ellipse.

00:04 This ellipse is described by the equation x squared divided by a square plus y square divided by b square is equal to 1.

00:10 And we want to prove that the line x is equal to a square divided by c is a directories for this ellipse.

00:17 To do so, we're going to prove that the line segment pf, which is this one right here, this one, divided by the line segment pd, which is just this one, is equal to the constant c over a.

00:33 A.

00:35 To start off, we're going to take the following equation for an ellipse in polar equation, in polar coordinates, which is r is equal to e times k divided by 1 plus e cosine theta.

00:48 And for an angle of theta is equal to zero, we'll have the following equation.

00:54 R is equal to e times k, divided by 1 plus e, since the cosine of theta is just zero.

01:01 And that is equal to a minus c.

01:05 So remember that the deal.

01:07 Distance from the center to this vertex over here is a.

01:15 The distance from the center to one of its foci, it's c.

01:20 And so the radius, in this case, just going to be the difference between a and c, which is this segment right here.

01:27 This is r.

01:29 So now we're just going to solve for, we're going to take this equation and solve for this constant right here.

01:35 So we'll have e times k is equal to a minus c times 1 plus e and so we also know that the constant e is equal to c divided by a so now i'm going to start working on this new whiteboard so we'll have e times k is equal to a minus c times 1 plus e we also know that e is equal to c divided by a so now we'll have the line segment pf which is this one right here, pf.

02:15 And so for this line segment, we have that the distance, the x distance from the foci to the point, is just going to be r -cosin theta.

02:29 And so for this point right here, we'll have that our radius over here, just going to be e times k divided by 1 plus e -cosin theta.

02:40 Well, we have stated that e times k is just equal to e a minus c times 1 plus e.

02:49 This is just going to be the same.

02:51 The denominator is the same.

02:54 1 plus e times cosine theta.

02:57 But we also have stated that e is c divided by a.

03:00 So we'll have a minus c plus 1 plus c divided by a divided by 1 plus c times a cosine and so now i'm going to start working with jal jorah over here.

03:18 So we'll have this same line sediment, which is just going to be a minus c.

03:24 I'm going to distribute this term right here.

03:28 I'm going to multiply, actually, i'm going to multiply everything through.

03:32 So we'll have a times 1, so it's just 1, plus c divided by a times a is just c.

03:39 So negative c times 1 is just negative c divided by a is just negative c squared divided by a.

03:50 So now the denominator stays i want to multiply everything through by an a over here.

03:55 So i'm going to multiply everything by a on the top and on the bottom.

04:01 So we'll have 1 plus c divided by a, cosine theta.

04:09 So we'll have this a is just going to become a square.

04:13 So these two c terms cancel each other minus c square, since i have multiplied by an a term.

04:21 So over here we'll have a plus c cosine theta.

04:29 And so now that we have this term right here, we have our we have a relationship for b square for our in ellipse which is b square is equal to a square minus c squared and this is exactly what we have over here this is the same so we can say that this term right here is going to be b square divided by a times cosine a plus c times cosine theta so we're going to stop right here and we have determined an equation for the line segment p now we're going to work our way through the segment, line segment, pd.

05:16 So in this case, pd, just going to be the distance between the x coordinate, which is x is equal to a squared divided by c, and our location for our point d.

05:28 So the x coordinate for our line segment is going to be a square divided by c, and our point is just located from this, from the origin.

05:43 Located a distance of c, sorry, c plus r -cosin theta.

05:56 So we have our distance from the center, which is our point of reference.

05:59 This is going to be the c part plus our r -cosin -theta, which depends on the angle that the point makes with respect to the foci.

06:12 And so if we distribute this negative sign, we'll have a square divided by c minus c minus r -cosin theta.

06:18 And so this term, we're just going to say that we're just going to multiply everything through because we already have determined...

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