1st PUC Mathematics · Chapter 11
Conic Sections — One Cone, Four Curves, and How to Tell Them Apart Instantly
The name “conic section” comes from a genuinely elegant fact: a circle, a parabola, an ellipse and a hyperbola are all produced by slicing a double cone at different angles. The exam doesn’t test the slicing, but understanding it makes the four standard equations feel like one family instead of four things to memorise separately.
Circle — The Simplest Case
A circle is the set of all points equidistant from a fixed centre. The standard equation with centre (h, k) and radius r is (x−h)² + (y−k)² = r². Centred at the origin, this simplifies to x² + y² = r².
Worked example: Find the equation of a circle with centre (2, −3) and radius 5. Directly substituting: (x−2)² + (y+3)² = 25.
Parabola — Distance to a Point Equals Distance to a Line
A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The standard form y² = 4ax opens rightward, with focus at (a, 0) and directrix x = −a. Three sibling forms follow the same pattern: y² = −4ax opens left, x² = 4ay opens up, x² = −4ay opens down — the sign and which variable is squared tells you the orientation instantly.
Worked example: Find the focus and directrix of y² = 12x. Comparing to y² = 4ax: 4a = 12, so a = 3. Focus is (3, 0); directrix is x = −3.
Ellipse — Sum of Distances Is Constant
An ellipse is the set of points where the sum of distances to two fixed points (the foci) stays constant. Standard form: x²/a² + y²/b² = 1. When a > b, the major axis lies along the x-axis; when b > a, it lies along the y-axis — a detail that decides which formula applies for the foci location, so always check which denominator is larger before proceeding.
The relationship c² = a² − b² (where c is the distance from centre to each focus, and a is the larger semi-axis) connects the shape to its foci — this single formula answers most “find the foci” questions once a and b are known.
Hyperbola — Difference of Distances Is Constant
A hyperbola is the set of points where the difference of distances to two foci stays constant. Standard form: x²/a² − y²/b² = 1. The relationship here flips slightly from the ellipse: c² = a² + b² instead of a² − b² — easy to confuse under exam pressure since the equations look so similar.
Recognising Which Curve You’re Looking At
Given any equation, three quick checks identify the conic: Are both x² and y² present with the same coefficient and sign? Circle. Is only one variable squared? Parabola. Are both squared with the same sign but different coefficients? Ellipse. Are both squared with opposite signs? Hyperbola.
Where Students Actually Lose Marks
The most common error is swapping the ellipse and hyperbola foci formulas — writing c² = a² − b² for a hyperbola instead of c² = a² + b². The second is forgetting to check which of a or b is larger in an ellipse before applying the major-axis formula, which flips the whole answer.
Frequently Asked Questions
Is a circle a special case of an ellipse?
Yes — when a = b in the ellipse equation, it becomes a circle with radius a. This is why the circle is sometimes described as an ellipse with zero eccentricity.
Why do parabola, ellipse and hyperbola all involve a focus?
Because all three are defined by a distance relationship to fixed point(s) — it’s the specific relationship (equal, constant sum, constant difference) that determines which of the three curves results.
Does this chapter connect to 3D Geometry later?
Indirectly — Introduction to Three Dimensional Geometry extends the same coordinate thinking into three dimensions, though conics themselves stay two-dimensional throughout 1st and 2nd PUC.
See the Pattern, Not Four Separate Chapters
Conic Sections rewards students who see the underlying family resemblance. Daniel Sir teaches it that way from the first class. See the PUC Mathematics Programme →