Coordinate GeometryHigh Yield (2 Qs in JEE)

Hyperbola & Rectangular Hyperbola Formulas

Standard hyperbola x²/a² - y²/b² = 1, asymptotes, conjugate hyperbola, rectangular hyperbola xy = c².

7 Formulas•320+ Questions in Bank•

Concept Diagram

Standard Parabola & Conic Geometry

Standard parabola y² = 4ax with focus S(a,0), directrix x = -a, vertex (0,0), and latus rectum length 4a.

e = \sqrt{1 + \frac{b^2}{a^2}}, \quad \text{Foci: } (\pm ae, 0), \quad \text{Directrices: } x = \pm \frac{a}{e}

Standard horizontal hyperbola opening along x-axis.

Examiner Tip: Eccentricity e is strictly greater than 1.

|SP - S'P| = 2a

Difference of distances from any point on hyperbola to the two foci is constant 2a.

Examiner Tip: Locus definition: |z - z₁| - |z - z₂| = 2a represents hyperbola.

y = mx \pm \sqrt{a^2 m^2 - b^2}

Condition of tangency: c² = a² m² - b².

Examiner Tip: Real tangents exist only when m² > b²/a².

y = \pm \frac{b}{a} x \iff \frac{x^2}{a^2} - \frac{y^2}{b^2} = 0

Straight lines passing through the origin asymptotic to the hyperbola branches.

Examiner Tip: Angle between asymptotes is 2 tan⁻¹(b/a) = 2 sec⁻¹(e).

xy = c^2, \quad \text{Parametric: } \left(ct, \frac{c}{t}\right), \quad e = \sqrt{2}

Asymptotes are mutually perpendicular coordinate axes.

Examiner Tip: Tangent at parameter t is x + y t² = 2ct; normal is y - c/t = t²(x - ct) or x t³ - y t - ct⁴ + c = 0.

-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \implies \frac{1}{e_1^2} + \frac{1}{e_2^2} = 1

Hyperbola whose transverse and conjugate axes are interchanged. Foci lie on y-axis at (0, ±be₂).

Examiner Tip: The hyperbola, its asymptotes, and conjugate hyperbola differ only by constant terms in their joint equations.

y = mx \mp \frac{m(a^2 + b^2)}{\sqrt{a^2 - b^2 m^2}}, \quad ax\cos\theta + by\cot\theta = a^2 + b^2

Normal to hyperbola with slope m and at parametric point (a sec θ, b tan θ).

Examiner Tip: Director Circle of Hyperbola is x² + y² = a² - b². Real director circle exists only if a > b.

Frequently tested misconceptions in Hyperbola & Rectangular Hyperbola