Conic SectionsJEE Main

Concept

Every conic section is the locus of points whose distance from a fixed focus is $e$ times the distance from a fixed directrix. That single number $e$ — the eccentricity — decides whether the curve is a circle, ellipse, parabola or hyperbola.

Key formula

Derivation

The focus-directrix definition $PF = e\cdot PD$ in polar form (focus at the pole) gives $r = l/(1 + e\cos\theta)$ where $l$ is the semi-latus rectum.

For $e<1$ the denominator never vanishes, so $r$ stays finite — a closed ellipse. At $e=1$ the curve opens into a parabola. For $e>1$ the denominator hits zero at the asymptotic directions, giving the two branches of a hyperbola.

Standard parameters follow: $a = l/|1-e^2|$, $c = ae$, and $b^2 = a^2|1-e^2|$.

Scenarios to explore

• Conic Sections — Parabola, ellipse & hyperbola from one family.
• Circle: e = 0 — Zero eccentricity makes the foci coincide.
• Parabola: e = 1 — The border case between closed and open conics.
• Hyperbola: e = 1.8 — Past 1 the curve opens into two branches.

Real-world applications

• Planetary and satellite orbits (ellipses; escape trajectories are hyperbolas).
• Projectile paths and suspension-bridge cables (parabolas).
• Reflecting telescopes and satellite dishes.

JEE exam tips

• Latus rectum length is $2b^2/a = 2l$ for ellipse and hyperbola, $4a$ for a parabola.
• Directrix is at $x = \pm a/e$; focus at $x = \pm ae$.

Common mistakes

• Thinking a circle has undefined eccentricity — it is simply $e = 0$.
• Using $b^2 = a^2(1-e^2)$ for a hyperbola (it needs $e^2 - 1$).
• Confusing the semi-latus rectum $l$ with the semi-major axis $a$.

Exam traps to avoid

• An ellipse has two foci and two directrices; a parabola has one of each.
• Eccentricity of a rectangular hyperbola is $\sqrt 2$.