JEE · Mathematics
Interpret a parabola through focus/directrix geometry, identify the correct standard orientation, and use focus, directrix, latus rectum, parametrisation, tangent and normal formulas with orientation-specific signs.
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In short
A parabola is the locus of points equidistant from a fixed focus and a fixed directrix, so its eccentricity is 1. For y²=4ax with a>0, the vertex is (0,0), the focus is (a,0), the directrix is x=-a, and the latus rectum length is 4a. Other orientations require corresponding sign and axis changes.
Do not present Advanced-only formula detail as explicit Main wording.
Vertex, axis, focus, directrix, latus rectum and parametric point must be located before any formula is applied.
Every parabola question starts by fixing the vertex, axis, focus, directrix, latus rectum and, where required, the parametric point, from the given standard-form equation.
Identify orientation, read 4a, place focus/directrix, select method, apply tangent/normal, validate equal distances.
Identify orientation, then read 4a, then place the focus and directrix, then select the coordinate or parameter method, then apply tangent or normal, then validate by checking equal focus/directrix distance.
There are four standard orientations for a>0, each with its own focus, directrix and axis.
Read orientation first. y²=4ax opens right with focus (a,0) and directrix x=-a; y²=-4ax opens left with focus (-a,0) and directrix x=a; x²=4ay opens up with focus (0,a) and directrix y=-a; x²=-4ay opens down with focus (0,-a) and directrix y=a. Use focus/directrix only after matching the form.
(at²,2at) with t∈R is Advanced-supported and valid only for y²=4ax.
Use (at²,2at) only for y²=4ax. Verify a point lies on the curve before applying a tangent formula, and transform signs and axes for other orientations rather than copying the same parameter form across orientations.
y²=12x gives 4a=12, so a=3. Vertex is (0,0), focus is (3,0), directrix is x=-3, and latus rectum length is 12.
P=(3,6) lies on the curve since 6²=36=12(3). Tangent at P: 6y=6(x+3), so y=x+3.
Validation: P satisfies the tangent equation, and the orientation is right-opening throughout.
e equals 1
Eccentricity of any nondegenerate parabola.
Use when — Confirming that a curve is a nondegenerate parabola by equal focus/directrix distance.
Latus rectum length equals 4a
Latus rectum length for y²=4ax, a>0.
Use when — a>0; endpoints (a,±2a) must satisfy the equation.
Point at t squared a, 2 a t
Parametric point on y²=4ax.
Use when — Advanced scope; t∈R; valid only for y²=4ax.
Common trap — Reusing this parameter form for a different orientation without transforming signs and axes.
y y sub 1 equals 2 a times x plus x sub 1
Tangent to y²=4ax at point P(x₁,y₁).
Use when — Advanced scope; P must lie on the parabola before use.
Common trap — Applying the tangent formula to a point that is not verified to be on the curve.
t y equals x plus a t squared
Tangent at the parametric point (at²,2at).
Use when — P=(at²,2at) on y²=4ax.
y equals negative t x plus 2 a t plus a t cubed
Normal at the parametric point (at²,2at).
Use when — Advanced scope; right-opening standard form y²=4ax.
Common trap — Using the slope check without confirming the standard right-opening form.
Reading a instead of 4a from the coefficient.
Execution errorWhy it happens
The coefficient in y²=4ax is 4a, not a, so misreading it produces a wrong focus and latus rectum.
How it is corrected
Divide the coefficient by 4 to isolate a before using any formula.
Using the wrong focus sign for the orientation.
Decision / selection errorWhy it happens
Each of the four standard orientations has a distinct focus sign; copying one orientation's sign onto another produces a wrong focus.
How it is corrected
Match the exact standard form to the orientation table before placing the focus.
Assuming every parabola opens right.
Knowledge gapWhy it happens
Sign and axis in the equation determine opening direction; it is not always y²=4ax.
How it is corrected
Identify orientation from the equation before applying any formula.
Using the tangent-at-point formula for a point not on the parabola.
Execution errorWhy it happens
The tangent formula yy₁=2a(x+x₁) is valid only when P lies on the curve.
How it is corrected
Substitute P into the parabola equation to confirm membership before using the tangent formula.
Transferring the (at²,2at) parameter or its tangent/normal formulas across orientations.
Decision / selection errorWhy it happens
The parametric point and its tangent/normal formulas are derived specifically for y²=4ax and do not transfer as-is.
How it is corrected
Rebuild the parametric point and tangent/normal for the actual orientation rather than reusing signs.
Presenting Advanced-only formula detail as explicit Main 2026 wording.
Knowledge gapWhy it happens
Main 2026 lists the standard parabola form but does not separately enumerate focus, directrix, eccentricity, parametric equations, tangent or normal.
How it is corrected
Check the scope-mapping table on this page before treating detailed formula content as Main scope.
FAQ
Straight answers about how Rank Sarthi fits into serious exam preparation.
Yes, in focused standard-form scope. Main 2026 explicitly includes the standard parabola form.
Yes. Advanced explicitly includes standard forms, focus, directrix, eccentricity, parametric equations and tangent/normal equations.
A parabola is defined as the locus of points equidistant from the focus and directrix, which fixes eccentricity at exactly 1 for any nondegenerate parabola.
Use only official Main and Advanced syllabus documents for scope claims and official archives for paper examples. Record orientation, focus, directrix, parameter, and validation for every worked item. Do not publish counts or trends.
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