Conic Sections

Written and reviewed by Devendra Dubey, Mathematics faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching Mathematics). Based on NCERT Mathematics Class XI, Chapter 10.

What shape does a torch make on the wall?

Hold a torch straight at a wall and the lit patch is a circle. Tilt it and the patch stretches into an oval. Tilt it further, and at one exact angle the oval springs open into a U-shaped curve that never closes. Tilt it more and the patch becomes the open curve of a hyperbola. The beam of a torch is a cone, the wall is a plane, and all four curves are what you get when a plane cuts a cone. These are the conic sections, and this chapter gives you the equation and the key numbers of each.

Slicing a cone

Rotate a line about a vertical axis, keeping a fixed angle $\alpha$ (the semi-vertical angle) between the line and the axis. This traces a double-napped cone: two cones joined at the vertex. Cut it with a plane that makes angle $\beta$ with the axis, not passing through the vertex.

If the plane passes through the vertex, the curves collapse. These are the degenerate conics.

Circle: all points at one distance from a centre

A circle with centre $(h,k)$ and radius $r$ is the set of points at distance $r$ from the centre. Writing the distance formula and squaring:

$$(x-h)^2 + (y-k)^2 = r^2$$

With the centre at the origin this becomes $x^2 + y^2 = r^2$. An equation with equal coefficients of $x^2$ and $y^2$ and no $xy$ term is a circle, and you find its centre and radius by completing the square.

Worked example: centre and radius. Find the centre and radius of $x^2 + y^2 + 8x + 10y - 8 = 0$.

Common wrong step: reading the centre as $(4,5)$, or the radius as $\sqrt 8$. The centre has the opposite signs of the numbers inside the brackets, and the radius is the square root of the right-hand side after completing the squares.

Parabola: equidistant from a point and a line

A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix) that does not contain the focus. The line through the focus perpendicular to the directrix is the axis; the parabola meets the axis at its vertex.

Why the equation is $y^2 = 4ax$. Put the vertex at the origin, the focus at $(a,0)$ and the directrix at $x = -a$. A point $(x,y)$ is on the parabola when its distance to the focus equals its distance to the directrix:

$$\sqrt{(x-a)^2 + y^2} = x + a \;\Longrightarrow\; (x-a)^2 + y^2 = (x+a)^2 \;\Longrightarrow\; y^2 = 4ax$$

Flip signs or swap $x$ and $y$ and you get the other three standard forms. The squared variable tells you the axis, and the sign of the other term tells you the direction.

The latus rectum is the chord through the focus perpendicular to the axis. For $y^2 = 4ax$, put $x=a$: $y^2 = 4a^2$, so $y=\pm 2a$ and the chord has length $4a$.

Worked example: reading a parabola. Find the focus, directrix and latus rectum of $x^2 = -12y$.

Common wrong step: taking $a = 12$. The coefficient is $4a$, so $a$ is one quarter of it.

Worked example: building a parabola. Find the equation of the parabola with vertex at the origin and directrix $y = 2$.

Common wrong step: writing $x^2 = 8y$. The curve always opens away from the directrix, so a directrix above the vertex means a downward parabola.

Ellipse: constant sum of distances

An ellipse is the set of points whose distances from two fixed points (the foci) add up to a constant $2a$, which must be larger than the distance $2c$ between the foci. A string pinned at two points and pulled tight by a pencil draws one.

The line through the foci is the major axis (length $2a$); the perpendicular bisector is the minor axis (length $2b$). The numbers $a$, $b$, $c$ are linked by a right triangle. Take the end $B$ of the minor axis. It is the same distance from both foci, and those two distances add to $2a$, so each equals $a$. The triangle with sides $b$, $c$ and hypotenuse $a$ then gives

$$c^2 = a^2 - b^2, \qquad e = \dfrac{c}{a} < 1$$

The eccentricity $e$ measures how stretched the ellipse is: near 0 is almost a circle, near 1 is very flat. It is below 1 because $c<a$.

The rule: the foci lie on the axis under the larger denominator. The latus rectum is the focal chord perpendicular to the major axis. Putting $x=c$ in the first form gives $y^2 = b^2\left(1 - \dfrac{c^2}{a^2}\right) = \dfrac{b^4}{a^2}$, so each half is $\dfrac{b^2}{a}$ and

$$\text{length of latus rectum} = \dfrac{2b^2}{a}$$

Worked example: a tilted-looking ellipse. For $9x^2 + 4y^2 = 36$, find the foci, eccentricity and latus rectum.

Common wrong step: taking $a^2 = 4$ because $x^2/4$ comes first. The letter $a$ always belongs to the larger denominator, wherever it sits.

Pause and predict: on which axis are the foci of $\dfrac{x^2}{16} + \dfrac{y^2}{25} = 1$, and how far from the centre?

Check your answer: 25 is the larger denominator and sits under $y^2$, so the foci are on the y-axis. Here $c = \sqrt{25-16} = 3$, so the foci are $(0,\pm3)$.

Pause and predict: can an ellipse have eccentricity exactly 1? What about exactly 0?

Check your answer: no to 1, and yes to 0. If $e=1$ then $c=a$, which makes $b=0$: the ellipse flattens into a line segment between the foci. If $e=0$ then $c=0$, the foci coincide at the centre and $a=b$, which is a circle.

Hyperbola: constant difference of distances

A hyperbola is the set of points whose distances from two fixed foci differ by a constant $2a$. The line through the foci is the transverse axis (length $2a$), the perpendicular through the centre is the conjugate axis (length $2b$), and the curve cuts the transverse axis at its two vertices.

The key relation looks like the ellipse's but with the opposite sign:

$$c^2 = a^2 + b^2, \qquad e = \dfrac{c}{a} > 1$$

Why $e>1$? Two sides of a triangle always differ by less than the third, so the constant difference $2a$ is less than the focal distance $2c$, which gives $c>a$.

The rule: the foci lie on the axis of the positive term. The latus rectum has the same formula as for the ellipse, $\dfrac{2b^2}{a}$ (put $x=c$ in the first form and $y^2 = \dfrac{b^4}{a^2}$ again). For $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ the lines $y = \pm\dfrac{b}{a}x$ are the asymptotes, the lines the branches approach but never meet. They are the diagonals of the rectangle of sides $2a$ and $2b$ drawn about the centre. A hyperbola with $a = b$ is equilateral and has $e = \sqrt2$.

Worked example: reading a hyperbola. Find the foci, eccentricity, latus rectum and asymptotes of $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$.

Common wrong step: using the ellipse relation $c^2 = a^2 - b^2 = 9 - 16 = -7$. A negative value of $c^2$ is impossible and signals the wrong formula.

Worked example: finding the equation. A hyperbola has foci $(0,\pm13)$ and a conjugate axis of length 24. Find its equation.

Common wrong step: computing $a^2 = c^2 + b^2$. For a hyperbola, $c$ is the largest of the three and $a^2 = c^2 - b^2$.

All four conics side by side

A quick way to name a conic from its equation: look at the coefficients of $x^2$ and $y^2$.

Beyond the textbook. For the exam: each conic has its own definition and its own formulas. In reality: there is one definition that covers the three non-circular conics. A conic is the set of points whose distance from a fixed point (the focus) is a constant multiple $e$ of its distance from a fixed line (the directrix). The value of $e$ alone decides the shape: less than 1 is an ellipse, exactly 1 is a parabola, more than 1 is a hyperbola. This is why the eccentricity column in the table above runs smoothly from 0 to 1 to beyond.

Beyond the textbook. For the exam: learn the formulas and the focus rules. In reality: the planets move around the Sun along ellipses with the Sun at one focus. The Earth's orbit has $e \approx 0.017$, so it is nearly a circle, while Halley's comet has $e \approx 0.97$, an extremely stretched ellipse. A body that passes the Sun once at high speed and never returns follows a hyperbola. A ball thrown in air, when air resistance is ignored, follows a parabola.

Beyond the textbook. For the exam: the focus of a parabola is a point on the axis at distance $a$ from the vertex. In reality: the focus is where a parabolic mirror collects all rays that arrive parallel to its axis. That is why satellite dishes and solar cookers place the receiver or pot at the focus, and why a headlamp with the bulb at the focus sends out a parallel beam. In an ellipse, a sound or light ray leaving one focus passes through the other focus, which is the principle of whispering galleries.

Exam traps

Formula sheet

60-second revision

Where this chapter connects

• Previous: Straight Lines supplies the distance formula and the coordinate methods that every conic derivation uses.
• Next: Introduction to Three Dimensional Geometry adds the third axis to the same coordinate ideas.
• Physics link: Gravitation shows planetary orbits as ellipses, and Motion in a Plane shows the parabolic path of a projectile.
• Class XII link: Application of Integrals finds areas bounded by circles, parabolas and ellipses.