Parabola & Classification of Conic Sections JEE Practice Flashcards

Vocabulary and formula-based flashcards for Class 12 Parabola and Conic Section classification, suitable for JEE Main and Advanced preparation.

Conic Section

The locus of a point that moves in a plane such that its distance from a fixed point (focus) bears a constant ratio to its perpendicular distance from a fixed line (directrix) which does not pass through the fixed point.

Focus (S)

The fixed point used in the definition of a conic section.

Directrix

The fixed line used in the definition of a conic section, which does not pass through the focus.

Eccentricity (e)

The constant ratio of the distance from the focus to the perpendicular distance from the directrix, denoted as $e = \frac{SP}{PM}$.

Axis

The line passing through the focus and perpendicular to the directrix.

Vertex

The point(s) where the conic section meets its axis.

Circle (Eccentricity)

A conic section where the eccentricity $e = 0$ (directrix is at infinity).

Parabola (Eccentricity)

A conic section where the eccentricity $e = 1$, meaning the distance from the focus equals the distance from the directrix.

Ellipse (Eccentricity)

A bounded, closed curve conic section where $0 < e < 1$.

Hyperbola (Eccentricity)

An unbounded conic section with two branches where $e > 1$.

Rectangular Hyperbola (Eccentricity)

A hyperbola where $e = \tan(x)$ is not applicable, but instead e = \root \times 2 (specifically \text{e} = \text{\root} 2) and asymptotes are mutually perpendicular.

Discriminant (Δ)

Determined as Δ = $abc + 2fgh - af^2 - bg^2 - ch^2$, used to identify if a second-degree equation represents a genuine or degenerate conic.

Non-degenerate Conic Condition

Represented by the condition Δ $\neq 0$.

Degenerate Conic Condition

Represented by the condition Δ = $0$, which may result in a point, a pair of parallel lines, or a pair of intersecting straight lines.

Parabola Condition (General Equation Method)

A non-degenerate conic where $h^2 - ab = 0$ and Δ $\neq 0$.

Circle Condition (General Equation Method)

A non-degenerate conic where $h^2 - ab < 0$, $a = b$, and $h = 0$.

Rectangular Hyperbola Condition (General Equation Method)

A non-degenerate conic where $h^2 - ab > 0$ and $a + b = 0$.

Focal Chord

Any chord of the parabola that passes through the focus.

Double Ordinate

A chord perpendicular to the axis of the parabola.

Latus Rectum (LR)

The specific double ordinate passing through the focus; its length in a standard parabola is $4a$.

Standard Form: y² = 4ax

A parabola opening right with vertex at $(0, 0)$, focus at $(a, 0)$, and directrix $x = -a$.

Focal Distance (y² = 4ax)

The distance of any point $P(x, y)$ from the focus, which equals $a + x$.

Focal Chord Property (t₁, t₂)

If a chord joining parametric points $t_1$ and $t_2$ on $y^2 = 4ax$ passes through the focus, then $t_1 t_2 = -1$.

Semi-Latus Rectum (Harmonic Mean)

The semi-latus rectum is the harmonic mean of the segments of any focal chord: $\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$.

Position of Point (S₁)

For a point $P(x_1, y_1)$ relative to $S \text{≡} y^2 - 4ax = 0$: $S_1 > 0$ is outside, $S_1 = 0$ is on, and $S_1 < 0$ is inside the parabola.

Condition of Tangency (Slope Form)

The line $y = mx + c$ is tangent to $y^2 = 4ax$ if $c = \frac{a}{m}$, where $m \neq 0$.

Point of Contact (Slope Form)

The coordinates where the tangent $y = mx + \frac{a}{m}$ touches the parabola: $(\frac{a}{m^2}, \frac{2a}{m})$.

T = 0 Method (Tangent Point Form)

The equation of the tangent at $(x_1, y_1)$ on $y^2 = 4ax$ given by $yy_1 = 2a(x + x_1)$.

Parametric Form of Tangent

The tangent at point 't' on $y^2 = 4ax$ is $ty = x + at^2$.

Pair of Tangents Equation

The combined equation of two tangents from an external point $P(x_1, y_1)$ given by $SS_1 = T^2$.

Chord of Contact

The line joining the two points of contact for tangents drawn from an external point, given by the equation $T = 0$.