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Your Class 11 chapter on conic sections opens with a name and two dates: Apollonius, 262 B.C. to 190 B.C. Most students skip past it. But the three words you use for these curves, parabola, ellipse and hyperbola, were his choice, and each one describes a precise property. Understanding why gives you a quick way to tell the three equations apart.
Who was Apollonius of Perga?
Very little is known about his life. His MacTutor biography gives his birth as about 262 BC in Perga, in what is now Antalya in Turkey, and his death as about 190 BC in Alexandria. As a young man he went to Alexandria, studied under followers of Euclid, and later taught there. Almost everything else we know comes from the prefaces to his books.
His great work, *Conics*, was written in eight books. Only the first four survive in Greek, while seven survive in Arabic translation. MacTutor quotes the historian T. L. Heath, who counted 387 separate propositions in the treatise. Apollonius also showed in *On the Burning Mirror* that a spherical mirror does not bring parallel rays to a single focus, and discussed the focal properties of a parabolic mirror, the idea behind today's dish antennas.
What were the conics called before Apollonius?
Heath's 1896 edition of the Treatise on Conic Sections explains that earlier geometers cut three different kinds of cone, each with a plane at right angles to one side. They named the curves after the cone: the "section of a right-angled cone" was the parabola, the "section of an acute-angled cone" the ellipse, and the "section of an obtuse-angled cone" the hyperbola. Archimedes still used these names. Apollonius instead derived all three curves from any cone, cut in the most general way.
Where do the names come from?
Heath traces the new names to a much older Greek method called the application of areas. He quotes a passage, reported by Proclus, describing "the application of areas, their exceeding, and their falling short" as discoveries of the Pythagoreans, "where we find the very terms afterwards applied by Apollonius to the three conic sections".
Parabola comes from the Greek for application: an area fitted exactly onto a line.
Ellipse comes from the Greek for falling short.
Hyperbola comes from the Greek for exceeding.
In modern algebra, put the vertex of each curve at the origin and call the length of its latus rectum p. Then the square on the ordinate, y², compares with the rectangle p × x like this:
Parabola: y² = px, exactly equal.
Ellipse: y² = px − (p/d)x², so y² falls short of px.
Hyperbola: y² = px + (p/d)x², so y² exceeds px.
Here d is the length of the major (or transverse) axis.
Worked example: checking the names with your own equations
Take the standard forms from the NCERT Class 11 chapter on conic sections.
Ellipse x²/25 + y²/9 = 1. Here a = 5, b = 3, and the latus rectum is 2b²/a = 18/5 = 3.6.
Move to the vertex (−5, 0) and measure x from there. A point 2 units from the vertex has original x = −3.
From the original equation: y² = 9(1 − 9/25) = 9 × 16/25 = 5.76.
Compare with p × x = 3.6 × 2 = 7.2. Then 5.76 is less than 7.2: it falls short, by 1.44.
Check: (p/d)x² = (3.6/10) × 4 = 1.44. The formula matches.
Hyperbola x²/16 − y²/9 = 1. Here a = 4, b = 3, and the latus rectum is 2b²/a = 18/4 = 4.5.
Measure from the vertex (4, 0). A point 1 unit beyond it has x = 5.
y² = 9(25/16 − 1) = 81/16 = 5.0625.
Compare with p × x = 4.5 × 1 = 4.5. Then 5.0625 exceeds 4.5, by 0.5625.
Check: (p/d)x² = (4.5/8) × 1 = 0.5625.
Parabola y² = 8x. Here p = 8. At x = 2, y² = 16 = 8 × 2 exactly.
You can also read the eccentricity from these: the ellipse has c = √(25 − 9) = 4, so e = 4/5 = 0.8, and the hyperbola has c = √(16 + 9) = 5, so e = 5/4 = 1.25.
Common mistakes with conic sections
Mixing up a and b. In an ellipse the larger denominator gives a. In a hyperbola a goes with the positive term.
Using c² = a² − b² for a hyperbola. For a hyperbola it is c² = a² + b².
Forgetting the latus rectum is 2b²/a. It is not b²/a, which is only half of it.
Conic sections are in the JEE (Main) 2026 syllabus, which lists sections of conics and their standard equations. For problem strategy, see our guide to approaching conic sections problems.
If you find it hard to picture how one cone gives three curves, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
What can a student learn from Apollonius?
Apollonius replaced three special constructions with one general one, and chose names that describe what the curves do rather than where they came from. When you learn a topic, look for the single idea that covers the special cases, and give each case a name you understand.
Practice routine:
For y² = 12x, x²/36 + y²/16 = 1 and x²/9 − y²/16 = 1, find the latus rectum and test one point each against px.
Write the eccentricity of each and say which is less than, equal to, or more than 1.
Explain to a friend, in two sentences, why the hyperbola is the curve that "exceeds".
Next step: do the three curves in step 1 and check that the falling-short and exceeding amounts match (p/d)x².
