JEE Maths›Conic Sections›Common Mistakes

Common Mistakes

Traps in Conic Sections

6 mistake patterns students fall for. 2 high-frequency traps appear in almost every exam.

Confusing eccentricity ranges for ellipse and hyperbola

Very CommonFORMULA

Ellipse has $0 < e < 1$ and hyperbola has $e > 1$ . Students often mix these up or forget the boundary cases.

Why: Both formulas involve $\sqrt{1 \pm b^2/a^2}$ and the sign difference is easy to overlook.

WRONG: Assuming

e > 1

for an ellipse, or

e < 1

for a hyperbola, leading to incorrect identification of the conic

RIGHT: Ellipse:

e = \sqrt{1 - b^2/a^2}

, so

0 < e < 1

. Hyperbola:

e = \sqrt{1 + b^2/a^2}

, so

e > 1

. Parabola:

e = 1

See pattern: Eccentricity and Related Problems →

Mixing up transverse and conjugate axes

Very CommonCASE MISS

The transverse axis is along the direction the hyperbola opens. Students confuse it with the conjugate axis, leading to wrong foci and vertex positions.

Why: For x²/a² - y²/b² = 1, transverse axis = 2a (along x), conjugate axis = 2b (along y). But students sometimes swap a and b.

WRONG: Placing foci along the conjugate axis, or computing vertices using b instead of a

RIGHT: Transverse axis has length 2a and contains the vertices and foci. For x²/a² - y²/b² = 1, it lies along the x-axis. Conjugate axis (2b) is perpendicular to it.

See pattern: Eccentricity and Related Problems →

Wrong sign in tangent condition for hyperbola

CommonSIGN ERROR

The tangency condition for hyperbola is $c^2 = a^2m^2 - b^2$ , but students often use the ellipse condition $c^2 = a^2m^2 + b^2$ .

Why: The ellipse and hyperbola tangency formulas differ only by a sign (+ vs -), and students default to the more familiar ellipse form.

WRONG: Using

c^2 = a^2m^2 + b^2

for a hyperbola tangent

RIGHT: For hyperbola:

c^2 = a^2m^2 - b^2

. For ellipse:

c^2 = a^2m^2 + b^2

. Remember: hyperbola subtracts, ellipse adds.

See pattern: Tangent and Normal Equations →

Forgetting both branches of the hyperbola

CommonCASE MISS

A hyperbola has two branches. Solutions that consider only one branch miss valid intersection points or tangent lines.

Why: Students visualize only the right branch of x²/a² - y²/b² = 1 and forget the left branch at x = -a sec theta.

WRONG: Finding only one tangent from an external point, or missing intersection points on the second branch

RIGHT: Always check both branches. When solving, ensure you consider both positive and negative values of sec theta (or both signs of x).

See pattern: Find Equation of Conic from Given Conditions →

Ignoring directrix orientation for horizontal vs vertical conics

CommonDOMAIN

When the axis of the conic is vertical (e.g., x² = 4ay), the directrix is horizontal (y = -a), not vertical.

Why: Students memorize directrix x = -a for y² = 4ax and apply the same form even when the parabola axis is vertical.

WRONG: Writing directrix as

x = -a

for the parabola

x^2 = 4ay

RIGHT: For

y^2 = 4ax

: directrix is

x = -a

(vertical axis along x). For

x^2 = 4ay

: directrix is

y = -a

(vertical axis along y). Match directrix orientation to the axis of symmetry.

See pattern: Find Equation of Conic from Given Conditions →

Using wrong parametric form for the conic

OccasionalFORMULA

Each conic has a different parametric form. Using the wrong one leads to incorrect tangent/normal equations.

Why: Students memorize parametric forms but mix them up under pressure. Parabola uses (at², 2at), ellipse uses (a cos t, b sin t), hyperbola uses (a sec t, b tan t).

WRONG: Using