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Ellipse questions combine three things: the shape numbers (a, b, c, e), a point on the curve, and a line that touches or is perpendicular to it. Students who work in x and y alone end up with long algebra. Students who use the eccentric angle usually need a few lines. This post explains where that angle comes from, derives the tangent and normal, and checks everything on one ellipse with real numbers.
Where is this on the syllabus?
The JEE (Advanced) 2026 brochure lists, for conics, "foci, directrices and eccentricity, parametric equations, equations of tangent and normal". The JEE (Main) 2026 syllabus lists conic sections "in standard forms". The current NCERT Class 11 chapter on conic sections defines eccentricity and the standard ellipse, but does not cover tangents, normals or the auxiliary circle. Check the documents for your exam year.
What are the shape numbers?
For x²/a² + y²/b² = 1 with a > b, NCERT gives c² = a² − b², where c is the distance from the centre to each focus, and defines the eccentricity as e = c/a. So the foci are (±ae, 0), and since c < a, e is between 0 and 1. Two more facts used in JEE:
The directrices are x = ±a/e.
For a point (x, y) on the ellipse, the distances to the two foci are a − ex and a + ex. They add to 2a, which is NCERT's definition of the ellipse.
What is the auxiliary circle?
Draw the circle x² + y² = a² on the major axis as diameter. Take a point Q = (a cos θ, a sin θ) on it, and drop a perpendicular to the x-axis. The point on the ellipse along that line is P = (a cos θ, b sin θ). It satisfies the equation, since cos²θ + sin²θ = 1. The angle θ is called the eccentric angle of P.
Every ordinate is shrunk by the same factor b/a. That is why the ellipse is a "squashed circle", and why results for the ellipse often come from the circle by replacing a sin θ with b sin θ.
How do you get the tangent and normal?
Tangent. The tangent at (x₁, y₁) is xx₁/a² + yy₁/b² = 1. At (a cos θ, b sin θ), this becomes
(x cos θ)/a + (y sin θ)/b = 1.
Slope form. The line y = mx + c touches the ellipse when c² = a²m² + b², so the tangents with slope m are y = mx ± √(a²m² + b²).
Normal. The tangent at θ has slope −(b cos θ)/(a sin θ), so the normal has slope (a sin θ)/(b cos θ). Through P, this simplifies to
ax/cos θ − by/sin θ = a² − b².
Worked example: x²/25 + y²/9 = 1
Shape numbers. a = 5, b = 3, c = √(25 − 9) = 4, so e = 4/5. Foci (±4, 0), directrices x = ±25/4.
A point. Take x = 4. Then y² = 9(1 − 16/25) = 81/25, so P = (4, 9/5), the top end of a latus rectum. Its eccentric angle has cos θ = 4/5 and sin θ = 3/5. The matching point on the auxiliary circle is Q = (4, 3), and 4² + 3² = 25, as it should be. The ordinate ratio is (9/5)/3 = 3/5 = b/a.
Focal distances. a − ex = 5 − (4/5)(4) = 1.8 and a + ex = 8.2. Directly: the distance to (4, 0) is 9/5 = 1.8, and to (−4, 0) it is √(64 + 81/25) = 41/5 = 8.2. They add to 10 = 2a.
Tangent. (x × 4/5)/5 + (y × 3/5)/3 = 1, so 4x/25 + y/5 = 1, or 4x + 5y = 25. Check at P: 16 + 9 = 25.
Normal. 5x/(4/5) − 3y/(3/5) = 25 − 9, so 25x/4 − 5y = 16, or 25x − 20y = 64. Check at P: 100 − 36 = 64. Its slope is 5/4, and the tangent's is −4/5; they multiply to −1.
Slope form. The tangents with slope 1 are y = x ± √(25 + 9) = x ± √34.
Which mistakes cost the most marks?
Using c² = a² + b², which is the hyperbola's relation.
Assuming a goes with x. If the larger denominator is under y², the major axis is vertical and the roles swap.
Treating θ as the angle at the centre to P.
Writing the normal with a² + b² on the right.
Not checking that a found point is actually on the ellipse.
How should you practise?
Sketch the ellipse, the auxiliary circle and the point before writing equations. Larkin and Simon (1987) argued that diagrams make visible information that equations leave implicit, which here is often which focus is nearer or which way a normal points.
Derive once a week: the tangent and normal at θ from the point form.
One ellipse, every result: pick a, b and θ, and check focal distances, tangent and normal numerically, as above.
Mix conics. In Rohrer, Dedrick and Stershic (2015), 126 seventh-grade students scored higher with interleaved practice one day and 30 days later. Our parabola guide gives you the other half of a mixed set.
If eccentricity or the directrix still feels abstract, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson.
Today, repeat the worked example for x²/16 + y²/7 = 1 at the point with x = 3 and y positive.
