MathsMediumClass 12

Auxiliary Circle, eccentric angle

Conic Sections

JEE Qs

Hard

min

Master the parametric forms and their geometric interpretation to efficiently solve complex problems involving tangents, normals, and loci on ellipses and hyperbolas.

🧮 Key Formulas

Ellipse: x^2/a^2 + y^2/b^2 = 1

Auxiliary Circle for Ellipse: x^2 + y^2 = a^2

Parametric coordinates of a point on Ellipse: P(a cosθ, b sinθ), where θ is the eccentric angle.

Hyperbola: x^2/a^2 - y^2/b^2 = 1

Auxiliary Circle for Hyperbola: x^2 + y^2 = a^2

Parametric coordinates of a point on Hyperbola: P(a secθ, b tanθ), where θ is the eccentric angle.

✅ Key Points for JEE

1The auxiliary circle for both ellipse (x^2/a^2 + y^2/b^2 = 1) and hyperbola (x^2/a^2 - y^2/b^2 = 1) is given by x^2 + y^2 = a^2, where 'a' is the semi-major/transverse axis length.
2For an ellipse, the eccentric angle θ of a point P(x,y) is geometrically defined by taking a point Q(a cosθ, a sinθ) on the auxiliary circle, such that P is obtained by scaling the y-coordinate of Q by b/a (i.e., P is the foot of the perpendicular from Q to the x-axis, then scaled vertically).
3The parametric forms (a cosθ, b sinθ) for ellipse and (a secθ, b tanθ) for hyperbola are extremely powerful for simplifying derivations of equations of tangents, normals, chords, and solving complex locus problems.
4Understanding the relationship between points on the conic and corresponding points on the auxiliary circle (especially for ellipse) can simplify geometric property derivations and problem-solving.

⚠️ Common Mistakes

✕Confusing the parametric forms for ellipse (a cosθ, b sinθ) and hyperbola (a secθ, b tanθ).
✕Incorrectly identifying the radius of the auxiliary circle as 'b' instead of 'a' for both conics.
✕Misinterpreting the geometric meaning of the eccentric angle for hyperbola, as it's not a direct polar angle of the point on the hyperbola itself but rather a parameter from a related construction.

📝 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 Mathematics Ch 11: Conic Sections