JEE (Main) Mathematics Conic Sections

Parabola Formulas

Equation of standard parabola :
\( y^2 = 4ax \), Vertex is \( (0, 0) \), focus is \( (a, 0) \), Directrix is \( x + a = 0 \) and Axis is \( y = 0 \)
Length of the latus rectum \( = 4a \), ends of the latus rectum are \( L(a, 2a) \) & \( L'(a, -2a) \).

Parametric Representation:
\( x = at^2 \) & \( y = 2at \)

Tangents to the Parabola \( y^2 = 4ax \):
1. Slope form \( y = mx + \frac{a}{m} (m \neq 0) \)
2. Parametric form \( ty = x + at^2 \)
3. Point form \( T = 0 \)

Normals to the parabola \( y^2 = 4ax \):
\( y - y_1 = -\frac{y_1}{2a} (x - x_1) \) at \( (x_1, y_1) \) ; \( y = mx - 2am - am^3 \) at \( (am^2, -2am) \) ; \( y + tx = 2at + at^3 \) at \( (at^2, 2at) \).

Ellipse Formulas

Standard Equation :
\( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a > b \) & \( b^2 = a^2 (1 - e^2) \).
Eccentricity: \( e = \sqrt{1 - \frac{b^2}{a^2}}, (0 < e < 1) \), Directrices : \( x = \pm \frac{a}{e} \).
Focii : \( S = (\pm ae, 0) \). Length of, major axes \( = 2a \) and minor axes \( = 2b \)
Vertices : \( A' \equiv (-a, 0) \) & \( A \equiv (a, 0) \).
Latus Rectum : \( \frac{2b^2}{a} = 2a(1 - e^2) \)

Auxiliary Circle :
\( x^2 + y^2 = a^2 \)

Parametric Representation :
\( x = a \cos \theta \) & \( y = b \sin \theta \)

Position of a Point w.r.t. an Ellipse:
The point \( P(x_1, y_1) \) lies outside, inside or on the ellipse according as ; \( \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} - 1 > < \text{ or } = 0 \).

Line and an Ellipse:
The line \( y = mx + c \) meets the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) in two points real, coincident or imaginary according as \( c^2 \) is \( < = \text{ or } > a^2m^2 + b^2 \).

Tangents:
Slope form: \( y = mx \pm \sqrt{a^2m^2 + b^2} \), Point form: \( \frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1 \),
Parametric form: \( \frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1 \)

Normals:
\( \frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2 - b^2 \), \( ax \sec \theta - by \text{ cosec } \theta = (a^2 - b^2) \), \( y = mx - \frac{(a^2 - b^2)m}{\sqrt{a^2 + b^2m^2}} \).

Director Circle:
\( x^2 + y^2 = a^2 + b^2 \)

HyperBola Formulas

Standard Equation:
Standard equation of the hyperbola is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), where \( b^2 = a^2 (e^2 - 1) \).
Focii : \( S \equiv (\pm ae, 0) \) Directrices : \( x = \pm \frac{a}{e} \)
Vertices : \( A = (\pm a, 0) \)
Latus Rectum (\( \ell \)) : \( \ell = \frac{2b^2}{a} = 2a (e^2 - 1) \).

Conjugate Hyperbola :
\( \frac{x^2}{a^2} - \frac{y^2}{b^2} = -1 \) & \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are conjugate hyperbolas of each.

Auxiliary Circle :
\( x^2 + y^2 = a^2 \).

Parametric Representation :
\( x = a \sec \theta \) & \( y = b \tan \theta \)

Position of A Point 'P' w.r.t. A Hyperbola :
\( S_1 = \frac{x_1^2}{a^2} - \frac{y_1^2}{b^2} - 1 > , = \text{ or } < 0 \) according as the point \( (x_1, y_1) \) lies inside, on or outside the curve.

Tangents :
(i) Slope Form : \( y = mx \pm \sqrt{a^2m^2 - b^2} \)
(ii) Point Form : at the point \( (x_1, y_1) \) is \( \frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1 \).
(iii) Parametric Form : \( \frac{x \sec \theta}{a} - \frac{y \tan \theta}{b} = 1 \).

Normals :
(a) at the point \( P(x_1, y_1) \) is \( \frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2 + b^2 = a^2e^2 \).
(b) at the point \( P(a \sec \theta, b \tan \theta) \) is \( \frac{ax}{\sec \theta} + \frac{by}{\tan \theta} = a^2 + b^2 = a^2e^2 \).
(c) Equation of normals in terms of its slope 'm' are \( y = mx \pm \frac{(a^2 + b^2)m}{\sqrt{a^2 - b^2m^2}} \).

Asymptotes :
\( \frac{x}{a} + \frac{y}{b} = 0 \) and \( \frac{x}{a} - \frac{y}{b} = 0 \). Pair of asymptotes : \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0 \).

Rectangular Or Equilateral Hyperbola :
\( xy = c^2 \), eccentricity is \( \sqrt{2} \).
Vertices : \( (\pm c \pm c) \); Focii : \( (\pm \sqrt{2}c, \pm \sqrt{2}c) \). Directrices : \( x + y = \pm \sqrt{2} c \)
Latus Rectum (\( \ell \)) : \( \ell = 2 \sqrt{2} c = \text{T.A.} = \text{C.A.} \)
Parametric equation \( x = ct, y = c/t, t \in R - \{0\} \)
Equation of the tangent at \( P(x_1, y_1) \) is \( \frac{x}{x_1} + \frac{y}{y_1} = 2 \) & at \( P(t) \) is \( \frac{x}{t} + ty = 2c \).
Equation of the normal at \( P(t) \) is \( xt^3 - yt = c(t^4 - 1) \).
Chord with a given middle point as \( (h, k) \) is \( kx + hy = 2hk \).