However You Slice It Conic Sections

Every curve in this chapter comes from the same simple setup: a double-napped cone, formed by spinning a slanted line around a fixed vertical line through a fixed point V, called the vertex. The vertex splits the double cone into two mirror-image halves called nappes, one above and one below. Now slice through this cone with a flat plane, and watch what happens as the tilt of that plane changes. Let alpha be the half-angle of the cone itself, and beta be the angle the cutting plane makes with the cone's central axis. When beta=90 degrees, the plane cuts straight across one nappe, giving a circle. Tilt it a little, so alpha<beta<90 degrees, and the cut becomes an oval: an ellipse. Tilt it further, until the plane runs exactly parallel to the cone's own slanted surface (beta=alpha), and the curve opens up permanently: a parabola. Tilt it past that point, so 0<=beta<alpha, and the plane slices through BOTH nappes at once, producing two separate mirror-image curves: a hyperbola. If the plane happens to pass exactly through the vertex V itself, the 'curve' collapses instead to a single point, a straight line, or a pair of crossing lines -- these degenerate cases sit right on the boundary between the real ones.

A circle is defined by the simplest possible distance rule: it is the set of every point in a plane that sits exactly the same distance from one fixed point. That fixed point is the centre, and the constant distance is the radius. Suppose the centre is C(h,k) and the radius is r, and let P(x,y) be any point on the circle. By definition, the distance CP must equal r, so applying the distance formula directly gives root[(x-h)^2 + (y-k)^2] = r. Squaring both sides removes the awkward square root, leaving the standard equation of a circle: (x-h)^2 + (y-k)^2 = r^2. When the centre happens to sit at the origin (h=k=0), this collapses to the familiar x^2+y^2 = r^2. Given any circle written in the expanded, messier form -- such as x^2+y^2+8x+10y-8=0 -- the centre and radius can always be recovered by completing the square on the x-terms and the y-terms separately, regrouping the equation back into (x-h)^2+(y-k)^2=r^2 form. That equation is worth memorising cold: it is quite literally the Pythagoras-style distance formula, squared, for a fixed distance r.

A parabola asks for a different kind of balance: instead of one fixed point, it needs a fixed point (the focus) AND a fixed line (the directrix), and it collects every point that is exactly equidistant from both. Placing the vertex at the origin, with focus F(a,0) sitting a>0 units to the right and directrix the vertical line x=-a sitting the same distance to the left, let P(x,y) be a general point on the curve. The condition PF = PB (where B is the foot of the perpendicular from P to the directrix) becomes, after applying the distance formula twice and simplifying, the remarkably clean equation y^2 = 4ax. Every point satisfying this equation really is equidistant from focus and directrix, and the reverse holds too. Because the equation only ever produces x>=0 (since y^2 can never be negative), the curve opens rightward from the vertex and extends forever, never closing up the way a circle or ellipse does. This asymmetry between one squared variable and one plain variable is the fingerprint of every parabola: whichever variable appears squared marks the axis of symmetry, and the sign of the other term decides which way the curve opens.

Placing the vertex at the origin, a parabola can open in exactly four possible directions, depending on which variable is squared and what sign its partner carries. y^2=4ax opens rightward (focus on the positive x-axis); y^2=-4ax opens leftward; x^2=4ay opens upward (focus on the positive y-axis); and x^2=-4ay opens downward. Spotting the direction takes only two checks: if y is the squared variable, the axis of symmetry is the x-axis, and if x is squared, the axis is the y-axis; then a positive coefficient opens the curve toward the positive direction of that axis, and a negative coefficient flips it. Every parabola also carries a special focal chord called the latus rectum: the chord through the focus, perpendicular to the axis, with both endpoints on the curve itself. For y^2=4ax, substituting x=a into the equation gives y^2=4a^2, so y=+-2a, meaning the latus rectum stretches from (a,2a) to (a,-2a) -- a length of exactly 4a. This single number, 4a, is often the fastest way to check or compare parabolas in an exam: read off a from the equation, and the latus rectum follows immediately, with no extra derivation needed.

An ellipse trades one focus for two: it is the set of every point whose distances to two fixed points, F1 and F2 (the foci), always add up to the same constant total. That constant sum is always written as 2a, and it must be larger than the distance between the two foci themselves, otherwise no such point could exist at all. The midpoint of F1F2 is the ellipse's centre; the axis running through both foci is the major axis, of length 2a; and the axis through the centre, perpendicular to it, is the minor axis, of length 2b. Placing the foci at (-c,0) and (c,0) and working through the sum condition PF1+PF2=2a using the distance formula -- a genuinely lengthy piece of algebra involving two rounds of squaring -- eventually simplifies down to the strikingly tidy standard equation x^2/a^2 + y^2/b^2 = 1. Checking a point at either end of the minor axis against this same sum condition reveals a beautifully clean relationship between all three lengths: a^2 = b^2 + c^2. A garden ellipse can even be drawn by hand using exactly this definition: loop a string loosely around two pins (the foci) and a pencil, then keep the string taut while tracing all the way around -- the total string length on either side of the pencil is fixed, so the sum of the two focus-distances stays constant by construction.

Two more numbers finish the picture of any ellipse. The eccentricity e=c/a measures how squashed the ellipse is compared to a perfect circle: since 0<c<a always holds for a genuine ellipse, e is always strictly between 0 and 1 -- close to 0 means nearly circular, close to 1 means very flattened. The latus rectum, exactly as with the parabola, is the focal chord perpendicular to the major axis passing through a focus; working out its half-length at the point directly above a focus gives length 2b^2/a. Take the ellipse x^2/25+y^2/9=1 as a worked check: here a^2=25 and b^2=9, so a=5, b=3, and c=root(a^2-b^2)=root(25-9)=root(16)=4. The foci sit at (+-4,0), the eccentricity is e=4/5, and the latus rectum measures 2(9)/5 = 18/5. Notice that the SAME denominator comparison rule that locates the major axis also applies in reverse: whichever variable's denominator is larger tells you where the major axis runs, so an equation like x^2/9+y^2/25=1 would instead have its major axis running along the y-axis, with the foci on the y-axis too.

A hyperbola looks at the same two-foci setup as an ellipse, but asks for a difference instead of a sum: it collects every point where the distance to the farther focus MINUS the distance to the closer focus stays fixed at the same constant, again written 2a. This time 2a must be SMALLER than the distance between the foci (2c), so c>a always for a hyperbola -- the reverse inequality from the ellipse. Working through PF1-PF2=2a with the foci at (-c,0) and (c,0) leads, after the same style of repeated squaring, to the standard equation x^2/a^2 - y^2/b^2 = 1, where b is defined by b^2=c^2-a^2 (the same three letters as the ellipse, but now paired by SUBTRACTION rather than addition). Because x^2/a^2 must be at least 1 for this equation to hold (since y^2/b^2 can never be negative), x is forced to satisfy x<=-a or x>=a -- there is a permanent gap between -a and a where no point of the curve ever appears, which is exactly why a hyperbola always splits into two separate, mirror-image branches rather than forming one connected curve the way an ellipse does. The eccentricity, still defined as e=c/a, is now always greater than 1, since c>a always holds here -- a clean, memorable contrast with the ellipse's 0<e<1.

These four curves are not just an algebra exercise -- they show up constantly wherever nature or engineering needs a very particular shape. A parabola has a genuinely useful optical property: any ray travelling parallel to its axis, after bouncing off the curve, passes through the focus exactly (and the reverse is true too: light placed at the focus reflects out in a perfectly parallel beam). That single fact is why torch and car headlight reflectors, satellite dishes, and telescope mirrors are all shaped as parabolas, with the bulb or receiver sitting deliberately at the focus. A parabolic mirror just 5cm deep with its focus 5cm from the vertex, for instance, works out to have an opening 60cm across -- a calculation that drops straight out of y^2=4ax once the depth is treated as an x-value. Suspension bridges tell a related but different story: a cable that hangs freely under its own weight forms a curve called a catenary, but once it carries a uniformly spread load along a horizontal roadway (via many vertical support wires), it settles into a genuine parabola instead. Ellipses appear whenever two distances must balance: every planet's orbit around the Sun is an ellipse with the Sun sitting at one focus, never at the centre, and whispering galleries exploit the same two-foci idea acoustically, since a sound made at one focus reflects around the curved ceiling and arrives clearly at the other focus, even in a noisy room.