Q1. Find the focus and directrix of the parabola y² = 12x.
Compare with y² = 4ax: 4a = 12 ⇒ a = 3
Focus = (a, 0) = (3, 0)
Directrix: x = −a = −3
Conic Sections — Class 11
Slice a cone at different angles and you get a circle, ellipse, parabola or hyperbola — each with one clean equation.
Circlex² + y² = r², centred at the origin with radius r.
Parabolay² = 4ax opens rightward, with focus at (a, 0) and directrix x = −a.
Ellipsex²/a² + y²/b² = 1 (a > b): a "squashed circle" with two foci.
Hyperbolax²/a² − y²/b² = 1: two separate curved branches opening left and right.
Solved examples
Q2. Find the equation of a circle with centre at the origin and radius 5.
x² + y² = r²
x² + y² = 5²
x² + y² = 25
Formula sheet
| Quantity | Formula |
|---|---|
| Circle | x² + y² = r² |
| Parabola | y² = 4ax |
| Ellipse | x²/a² + y²/b² = 1 |
| Hyperbola | x²/a² − y²/b² = 1 |
Common mistakes & exam wins
- All conics come from slicing a double cone — hence the name "conic sections".
- For an ellipse x²/a² + y²/b² = 1, the LARGER denominator tells you which axis is the major axis.
- A circle is just a special ellipse where a = b.
Frequently asked questions
What is the standard equation of a parabola?
y² = 4ax, with vertex at the origin, focus at (a, 0), and directrix x = −a.
What is the equation of an ellipse?
x²/a² + y²/b² = 1, centred at the origin, with semi-axes a and b.
What are the four types of conic sections?
Circle, parabola, ellipse and hyperbola — each obtained by slicing a cone at a different angle.