JEE · Mathematics
Interpret a nondegenerate ellipse, distinguish horizontal and vertical major-axis forms, calculate foci and eccentricity, and use directrix, latus rectum, parametric, tangent and normal results under correct orientation and parameter conditions.
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In short
For x²/a²+y²/b²=1 with a>b>0, the semi-major axis is a along x, the semi-minor axis is b, c²=a²-b², foci are (±c,0), and e=c/a, so 0<e<1. The full major-axis length is 2a, not a.
Do not present Advanced-only formula detail as explicit Main wording.
Semi-major axis, semi-minor axis, focal distance c, eccentricity, directrix and latus rectum must be located before any formula is applied.
Every ellipse question starts by fixing the semi-major axis, semi-minor axis, focal distance c, eccentricity and, where required, the directrix and latus rectum, from the given standard-form equation.
Standardise, identify the larger denominator, assign a and b, compute c², then e, then method, then validate 0<e<1.
Standardise the equation, identify the larger denominator, assign a and b accordingly, compute c², then e, then select the focal, parameter or tangent method, then validate by checking 0<e<1.
The larger denominator gives the major-axis direction; the vertical-major form uses a different foci placement.
For a horizontal-major ellipse, x²/a²+y²/b²=1 with a>b>0 has foci (±c,0) and directrices x=±a/e. For a vertical-major ellipse, x²/b²+y²/a²=1 with a>b>0 has foci (0,±c) and directrices y=±a/e. The larger denominator always identifies the major-axis direction.
a and b are semi-axis lengths; the full major and minor axis lengths are 2a and 2b.
a is the semi-major axis, not the full major-axis length. The full major-axis length is 2a, and the full minor-axis length is 2b. Confusing a with the full axis length is a common source of error.
For x²/25+y²/9=1, a=5, b=3, so c²=25-9=16, c=4.
Foci are (±4,0), e=4/5, directrices are x=±25/4 in Advanced scope, and the major-axis length is 10.
Validation: 0<4/5<1, c<a, and the larger denominator sits under x², confirming the horizontal-major orientation.
c squared equals a squared minus b squared
Relation between semi-major axis, semi-minor axis and focal distance.
Use when — a>b>0; result satisfies 0<c<a.
e equals c over a
Eccentricity of the ellipse.
Use when — Nondegenerate ellipse; result satisfies 0<e<1.
x equals plus or minus a over e, equals plus or minus a squared over c
Directrices for the horizontal-major standard ellipse.
Use when — Advanced scope; c>0; directrices lie outside the vertices.
Latus rectum length equals 2 b squared over a
Latus rectum length of the ellipse.
Use when — Nondegenerate ellipse.
Point at a cos theta, b sin theta
Parametric point on the horizontal-major ellipse.
Use when — Advanced scope.
x x sub 1 over a squared plus y y sub 1 over b squared equals 1
Tangent at point P(x₁,y₁) on the ellipse.
Use when — Advanced scope; P must lie on the ellipse before use.
Common trap — Applying the tangent formula to a point not verified to be on the curve.
x cos theta over a plus y sin theta over b equals 1
Tangent at the parametric point.
Use when — Parametric P on the ellipse.
a x sin theta minus b y cos theta equals a squared minus b squared, times sin theta cos theta
Normal at the parametric point.
Use when — Advanced scope; horizontal-major form.
Common trap — Not checking axis-endpoint cases separately.
Treating a as the full major-axis length instead of the semi-major axis.
Knowledge gapWhy it happens
a is a semi-axis length; the full major-axis length is 2a.
How it is corrected
State a as semi-major axis and 2a as full major-axis length explicitly.
Using the hyperbola focal relation c²=a²+b² for an ellipse.
Recall gapWhy it happens
Ellipse and hyperbola have different focal relations; the ellipse relation is c²=a²-b².
How it is corrected
Confirm the conic type before applying a focal relation formula.
Assigning the larger denominator to b instead of a.
Decision / selection errorWhy it happens
By convention a>b>0, and the larger denominator identifies the major-axis direction.
How it is corrected
Compare denominators first and assign a to the larger one before proceeding.
Computing e outside 0<e<1 for a claimed ellipse.
Execution errorWhy it happens
A valid nondegenerate ellipse always has eccentricity strictly between 0 and 1.
How it is corrected
Recheck the focal relation and denominators if computed e falls outside this range.
Placing foci on the wrong axis for the given orientation.
Decision / selection errorWhy it happens
Horizontal-major and vertical-major ellipses place foci on different axes.
How it is corrected
Match orientation to the larger-denominator axis before placing foci.
Applying the tangent formula before verifying the point lies on the ellipse.
Execution errorWhy it happens
The tangent-at-point formula is valid only for points on the curve.
How it is corrected
Substitute the point into the ellipse equation before using the tangent formula.
FAQ
Straight answers about how Rank Sarthi fits into serious exam preparation.
Yes, in focused standard-form scope. Main 2026 explicitly includes the standard ellipse form.
Yes. Advanced explicitly includes standard form, foci, directrices, eccentricity, parametric equations and tangent/normal equations.
Because c²=a²-b² with a>b>0 gives 0<c<a, and e=c/a, so e always lies strictly between 0 and 1 for a nondegenerate ellipse.
Use only official Main and Advanced syllabus documents for scope claims and official archives for paper examples. Record orientation, a, b, c and validation for every worked item. Do not publish counts or trends.
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