Ellipse Deep DiveJEE Main

Two foci, one constant sum — interactive Mathematics simulation for IIT-JEE.

Concept

An ellipse is all points whose distances to two foci sum to a constant (2a) — the pins-and-string construction. Shape is measured by eccentricity $e = c/a \in (0,1)$ : near 0 is circular, near 1 is cigar-like. The eccentric angle parametrises it via the auxiliary circle: (a·cosφ, b·sinφ).

Key formula

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1; \quad b^2 = a^2(1-e^2); \quad PF_1 + PF_2 = 2a; \quad \ell = \frac{2b^2}{a}

Derivation

Sum-of-distances definition + algebra (square twice) delivers the standard equation with $c^2 = a^2 - b^2$ .

Focal distances individually: $r_{1,2} = a \pm ex$ — linear in x! The parametric point comes from squashing the auxiliary circle x² + y² = a² vertically by b/a.

Scenarios to explore

Ellipse Deep Dive — Two foci, constant sum — eccentric angle & auxiliary circle.

Real-world applications

Planetary orbits (Kepler I) — the Sun at one focus.
Whispering galleries & lithotripsy: rays from one focus converge at the other.
Elliptical gears & cams.

JEE exam tips

Focal distances a ± ex solve most 'find PF' questions without square roots.
Director circle x² + y² = a² + b²: from any point on it the two tangents are perpendicular.
Area πab; perimeter has NO elementary form (Ramanujan approximations only).

Common mistakes

b² = a² − c² sign confusion with the hyperbola's b² = c² − a².
Eccentric angle φ is NOT the polar angle of the point.
Latus rectum 2b²/a — forgetting the 2.

Exam traps to avoid

If b > a in the given equation, the major axis is VERTICAL — foci on the y-axis.
e of a circle is 0; JEE options bait with e = 1 (that's a parabola).