MathsMediumClass 12

Rectangular Hyperbola — xy = c²

Conic Sections

JEE Qs

Hard

min

Master the parametric form `(ct, c/t)` as it is the most efficient tool for solving problems related to tangents, normals, and chords of `xy = c^2`.

🧮 Key Formulas

Equation of Rectangular Hyperbola: xy = c^2

Parametric form: (ct, c/t)

Eccentricity: e = sqrt(2)

Asymptotes: x = 0, y = 0 (the coordinate axes)

Equation of Tangent at (x1, y1): xy1 + yx1 = 2c^2 or x/x1 + y/y1 = 2

Equation of Tangent at (ct, c/t): x/t + yt = 2c

Equation of Normal at (ct, c/t): xt^3 - yt - c(t^4 - 1) = 0

Equation of Chord joining P(ct1, c/t1) and Q(ct2, c/t2): x + t1t2y = c(t1 + t2)

✅ Key Points for JEE

1A rectangular hyperbola is a hyperbola whose asymptotes are perpendicular. The equation `xy = c^2` represents such a hyperbola with its asymptotes along the coordinate axes.
2The eccentricity of a rectangular hyperbola is always `sqrt(2)`.
3The parametric form `P(ct, c/t)` is exceptionally useful for deriving equations of tangents, normals, and chords, simplifying complex calculations.
4The area of the triangle formed by any tangent to the rectangular hyperbola `xy = c^2` and its asymptotes (x=0, y=0) is a constant, equal to `2c^2`.
5The product of the perpendicular distances from any point on the rectangular hyperbola `xy = c^2` to its asymptotes (x=0, y=0) is a constant, `c^2`.

⚠️ Common Mistakes

✕Confusing the properties of `xy = c^2` with the standard form `x^2/a^2 - y^2/b^2 = 1` without considering the rotation of axes or the specific nature of rectangular hyperbola.
✕Errors in algebraic manipulation when using the parametric form `t` for calculating slopes, tangent, or normal equations.
✕Forgetting that the asymptotes for `xy=c^2` are simply the x and y axes, which significantly simplifies problems involving asymptotes.

📝 Practice Questions

See all

Q70.Let S = {(x, 1}, where (1) An ellipse whose eccentricity is 1 , when (2) A hyperbola whose eccentricity is 2 , when √r+1 √r+1 r > 1. 0 < r < 1. (3) (4) A hyperbola whose eccentricity is 2 , when An ellipse whose eccentricity is , when √1−r √ r+12 r > 1 0 < r < 1

2019·MCQHard

Q69.If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P , then the distance of P from the origin (units), is: + (1) 2(3 2√2) (2) 3 + 2√2 + (3) √2 + 1 (4) 2(√2 1) JEE Main 2017 (08 Apr Online) JEE Main Previous Year Paper

2017·Multi conceptHard

NCERT Chapters

Class 11 Maths Ch 11: Conic Sections