JEE · Mathematics
Identify hyperbola orientation, use transverse/conjugate axes, foci, eccentricity and asymptotes consistently with the standard form, and apply directrix, parameter, tangent and normal methods under Advanced-supported conditions.
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In short
For x²/a²-y²/b²=1 with a,b>0, the transverse axis is the x-axis, c²=a²+b², foci are (±c,0), and e=c/a>1. Its asymptotes are y=±(b/a)x. These relations must be rebuilt when orientation changes.
Do not present Advanced-only formula detail as explicit Main wording.
Transverse axis, conjugate axis, vertex, asymptote and eccentricity must be located before any formula is applied.
Every hyperbola question starts by fixing the transverse axis, conjugate axis, vertices, asymptotes and eccentricity from the given standard-form equation.
Standardise, locate the positive squared term, fix the transverse axis, compute c², then e/asymptotes, then method, then validate e>1.
Standardise the equation, locate the positive squared term, identify the transverse axis, compute c², then e and the asymptotes, then select the parameter or tangent method, then validate by checking e>1.
Opposite squared signs mean c²=a²+b², unlike the ellipse; asymptotes come from the highest-degree part of the equation.
Because the two squared terms carry opposite signs, the focal relation is c²=a²+b², not the ellipse's c²=a²-b². Asymptotes must be tied to the actual orientation of the standard form rather than copied from a memorised horizontal case.
At a vertex the tangent is vertical, so the normal there is found by geometry, not by the general nonvertex normal formula.
At a vertex (±a,0) the tangent is vertical, so the normal is the horizontal line y=0, found by geometry rather than the general nonvertex normal formula which requires y₁≠0.
For x²/9-y²/16=1, a=3, b=4, so c²=9+16=25, c=5, and e=5/3.
Foci are (±5,0), asymptotes are y=±4x/3. At vertex (3,0), the tangent is x=3 and the normal is y=0.
Validation: c>a and e>1, consistent with a nondegenerate hyperbola.
Vertices at plus or minus a, zero
Vertices of the horizontal-transverse hyperbola.
Use when — Horizontal transverse axis; vertices lie on the curve.
c squared equals a squared plus b squared
Relation between semi-transverse axis, semi-conjugate axis and focal distance.
Use when — a,b positive; result satisfies c>a.
Common trap — Using the ellipse relation c²=a²-b² instead.
e equals c over a
Eccentricity of the hyperbola.
Use when — Nondegenerate hyperbola; result satisfies e>1.
x equals plus or minus a over e, equals plus or minus a squared over c
Directrices for the horizontal-transverse standard hyperbola.
Use when — Advanced scope; correct focal ratio required.
Latus rectum length equals 2 b squared over a
Latus rectum length of the hyperbola.
Use when — Nondegenerate hyperbola.
y equals plus or minus b over a, times x
Asymptotes of the horizontal-transverse standard hyperbola.
Use when — This horizontal-transverse form; derived from the highest-degree part.
Common trap — Using the horizontal-form asymptote slope for a vertical-transverse hyperbola.
Point at a sec theta, b tan theta
Parametric point on the horizontal-transverse hyperbola.
Use when — Advanced scope; cosθ≠0.
Common trap — Omitting the cosθ≠0 domain restriction.
x x sub 1 over a squared minus y y sub 1 over b squared equals 1
Tangent at point P(x₁,y₁) on the hyperbola.
Use when — Advanced scope; P must lie on the curve before use.
x sec theta over a minus y tan theta over b equals 1
Tangent at the parametric point.
Use when — Valid parameter, cosθ≠0.
y minus y sub 1 equals negative, a squared y sub 1 over b squared x sub 1, times x minus x sub 1
Normal at a non-vertex point P on the hyperbola.
Use when — Advanced scope; y₁≠0; P on the standard hyperbola.
Common trap — Applying this formula at a vertex where y₁=0.
y equals zero
Normal at the vertex (±a,0) of the horizontal-transverse hyperbola.
Use when — Horizontal-transverse form; tangent at the vertex is vertical.
Using the ellipse focal relation c²=a²-b² for a hyperbola.
Decision / selection errorWhy it happens
The hyperbola's opposite squared signs give c²=a²+b², not the ellipse's c²=a²-b².
How it is corrected
Confirm the conic type from the sign pattern before applying a focal relation formula.
Computing e<1 for a claimed hyperbola.
Execution errorWhy it happens
A valid nondegenerate hyperbola always has eccentricity strictly greater than 1.
How it is corrected
Recheck the focal relation if computed e is not greater than 1.
Using the horizontal-form asymptote slope for a vertical-transverse hyperbola.
Decision / selection errorWhy it happens
Asymptote slope depends on which axis is the transverse axis.
How it is corrected
Derive asymptotes from the highest-degree part of the actual standard form given.
Confusing the transverse axis with the conjugate axis.
Knowledge gapWhy it happens
The transverse axis contains the vertices and foci; the conjugate axis does not.
How it is corrected
Identify the transverse axis from the positive squared term before naming axis lengths.
Applying the nonvertex normal formula at a vertex.
Execution errorWhy it happens
The nonvertex normal formula requires y₁≠0, which fails at a vertex where y₁=0.
How it is corrected
Use the geometric vertex-normal result y=0 at (±a,0) instead.
Omitting the cosθ≠0 condition for the parametric point.
Recall gapWhy it happens
The parametric point (a secθ, b tanθ) is undefined when cosθ=0.
How it is corrected
State cosθ≠0 explicitly whenever this parametric form is used.
FAQ
Straight answers about how Rank Sarthi fits into serious exam preparation.
Yes, in focused standard-form scope. Main 2026 explicitly includes the standard hyperbola form.
Yes. Advanced explicitly includes foci, directrices, eccentricity, parametric equations and tangent/normal equations.
Because c²=a²+b² with a,b>0 gives c>a, and e=c/a, so e is always greater than 1 for a nondegenerate hyperbola.
Use only official Main and Advanced syllabus documents for scope claims and official archives for paper examples. Record orientation, a, b, c, e and validation for every worked item. Do not publish counts or trends.
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