It can be shown that, in general, for any ellipse and any...

It can be shown that, in general, for any ellipse and any hyperbola, there are two straight lines called directrices, perpendicular to the focal axis and at distance $a / e=a^2 / c$ from the center, such that the equation of the curve can be derived from the relation $P F=e \cdot P D$, where $P F$ is the distance from a point on the curve to a focus and $P D$ is the perpendicular distance to the directrix. (See Figs. 40-2 and 40-3.) (FIGURE CAN'T COPY)

Key Concepts

Conic Sections

Conic sections are the curves formed by the intersection of a plane with a double-napped cone. The broad categories include ellipses, parabolas, and hyperbolas. Each type of conic section is defined using a fixed point called a focus (or foci) and a fixed line called a directrix, with their shapes determined by a parameter known as eccentricity.

Focus

A focus is a fixed point (or in some cases, points) in the plane that, together with a directrix, is used to define a conic section. The distance of any point on the conic from the focus plays a critical role in the curve’s construction and its geometric properties.

Directrix

A directrix is a fixed straight line that, in conjunction with a focus, sets the stage for the geometric definition of conic sections. The distance from any point on the conic to the directrix, measured perpendicularly, is used along with the distance to the focus to characterize the curve.

Eccentricity

Eccentricity is a nonnegative real number that quantifies the deviation of a conic section from being circular. It serves as a constant ratio in the distance definition of conics: for any point on the curve, the distance to the focus divided by the perpendicular distance to the directrix always equals the eccentricity. Its value determines the type of conic: ellipses (eccentricity less than 1), parabolas (eccentricity equal to 1), and hyperbolas (eccentricity greater than 1).

Locus Definition of Conics

The locus definition of conic sections states that a conic is the set of all points for which the ratio of the distance to the focus to the distance to the directrix is constant (equal to the eccentricity). This fundamental relationship, expressed as PF = e · PD, encapsulates the geometric essence of conic sections and underpins their derivation and classification.

Transcript

00:01 In this question, we have to show that each point p of the hyperbola, the tangent line bisects the angle between the focal radiae f1p and f2 p.

00:17 Now let hyperbola is x squared divided by a square minus y squared divided by b squared equals to 1.

00:33 Now we know that equation of tangent of hyperbola at point 5 is x by a multiplied by secant 5 minus y by b times y by b tangent 5 equals to 1.

00:58 Now let its gradient is m1 then m1 equals to minus coefficient of x which is minus 1 by a secant 5 divided by coefficient of y which is minus 1 by b tangent phi in a simplified form we can write it as b second 5 divided by a tangent 5 this we can write it as b by a times we can write secant phi as 1 by cosine phi and tangent 5 as sine 5 as sine 5 so it will be cosine 5 divided by sine pi.

02:11 In simplified form, we can write it as m1 equals to b by a times 1 by sine pi.

02:24 Now we know that the gradient of a line which passes through points x1 by 1 and x2 y2 is m equals to y2 minus y1 divided by x2 minus x1.

02:40 So gradient of flying f1p is m2 then m2 equals to b tangent phi minus 0 divided by a seccent phi minus ae.

03:13 So we can write this b tangent 5 divided by we will take common a secund phi minus e.

03:29 Now let represent it by b.

03:41 Now because here x1 y1 is ae.

03:45 0 and x2.

03:47 2 .2 is a secant 5 and b tangent 5 and gradient of line f2 p is m3, then m3 equals to b tangent 5 minus 0 divided by a secend 5 minus bracket minus ae.

04:18 In simplified form you can write it as b tangent 5 divided by a times secant 5 plus c let mark it as c.

04:40 Now because here x1 y1 is minus ae comma 0 and x2 y2 is is a secant d and b tangent b let tangent at p divided the angle between f1 p and f2 p is is alpha and beta...

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