Ellipse Lecture Practice Flashcards

Vocabulary and key formulas related to the geometry, properties, and equations of ellipses extracted from the lecture transcript.

Eccentricity ($e$)

The ratio of the distance from a fixed point ($S$) to the distance from a fixed line ($PM$), where the locus is an ellipse if $0 < e < 1$.

Standard form of Ellipse ($a > b$)

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $b^2 = a^2(1 - e^2)$.

Vertices

The points $A(a, 0)$ and $A'(-a, 0)$ on the major axis where it intersects the ellipse.

Auxillary circle

A circle described on the major axis of an ellipse as its diameter, represented by the equation $x^2 + y^2 = a^2$.

Eccentric angle ($\theta$)

The angle used to define parametric coordinates on an ellipse, where any point $P'$ on the auxillary circle corresponds to point $P$ on the ellipse.

Parametric form of Ellipse

Coordinates expressed as $(a \times \text{cos}(\theta), b \times \text{sin}(\theta))$.

Length of latus rectum ($L.L.R.$)

The length given by the formula $\frac{2b^2}{a}$ for an ellipse where $a > b$, or $\frac{2a^2}{b}$ where $b > a$.

Focal distance property ($SP + SP'$)

The sum of the distances from any point on the ellipse to the two foci is constant and equal to the length of the major axis ($2a$).

Position of a point ($S_{11}$)

A point $(x_1, y_1)$ is outside the ellipse if $S_{11} > 0$, on the ellipse if $S_{11} = 0$, and inside the ellipse if $S_{11} < 0$.

Condition for tangency

For a line $y = mx + c$ to be a tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the constant must satisfy $c^2 = a^2 m^2 + b^2$.

Director circle

The locus of the point of intersection of two perpendicular tangents, defined by the equation $x^2 + y^2 = a^2 + b^2$.

Harmonic mean of segments of focal chord

The value $\frac{2 \times SP \times SQ}{SP + SQ}$ which is equal to the semi-latus rectum $\frac{b^2}{a}$.

Product of perpendiculars from foci

The product of lengths of perpendiculars from the foci onto any tangent is equal to the square of the semi-minor axis ($d_1 \times d_2 = b^2$ for $a > b$).

Center of the ellipse ($a > b$)

The midpoint of the vertices $A$ and $A'$ or the midpoint of the foci $S$ and $S'$, located at $(0, 0)$ in standard form.