Conic Sections

Excerpts

• Conic Sections

  If $ab - h^{2}$ is a positive number, the curve is an ellipse; if $ab - h^{2} = 0$, the curve is a parabola: and if $ab - h^{2}$ is a negative number, the curve is a hyperbola.

• Conic Sections

  The characteristic property of a focus, $S$, and its corresponding directrix, $XN$, for any one of the three types of curve, is that the ratio $SP$ to $PN$ $\left(\ie\ \dfrac{SP}{PN}\right)$ is constant, where $PN$ is the perpendicular on the directrix from $P$, and $P$ is any point on the curve.

• Conic Sections

  The orbits of the planets are ellipses, the sun being in the focus.

• Conic Sections

  There is a certain type of mathematician who is always rather impatient at delaying over the ideas of a subject: he is anxious at once to get on to the proofs of “important” problems. The history of the science is entirely against him.

• Conic Sections

  Novel ideas are more apt to spring from an unusual assortment of knowledge---not necessarily from vast knowledge, but from a thorough conception of the methods and ideas of distinct lines of thought.

• Conic Sections

  Nothing illustrates better the gain in power which is obtained by the introduction of relevant ideas into a science than to observe the progressive shortening of proofs which accompanies the growth of richness in idea.

• Conic Sections

  No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature.

• Conic Sections

  There are accordingly three types of conic sections, namely, ellipses, parabolas, and hyperbolas.

• Conic Sections

  Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all three curves.

• Conic Sections

  (1) The orbits of the planets are ellipses, the sun being in the focus.

• Conic Sections

  This sweeping general law, coupled with the three laws of motion which he put into their final general shape, proved adequate to explain all astronomical phenomena, including Kepler’s laws, and has formed the basis of modern physics.

• Conic Sections

  This fact is worth noting; for it is characteristic of modern mathematics to include among general forms all sorts of particular cases which would formerly have received special treatment.

Equations

• Conic Sections

  \dfrac{SP}{PN}

  For a focus S and its corresponding directrix XN, the ratio of the distance SP to the perpendicular PN from any point P on the curve to the directrix is constant, for ellipses, parabolas and hyperbolas alike.

• Conic Sections

  ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0

  The general algebraic equation of the second degree, which when it represents any locus always represents a conic section, and to which the equation of every conic section can be brought.

• Conic Sections

  ab - h^{2} = 0

  When ab - h^2 is zero, the conic given by the general second-degree equation is a parabola.

• Conic Sections

  a(x^{2} + y^{2}) + 2gx + 2fy + c = 0

  The equation of any circle can be written in this form, with the coefficients of x^2 and y^2 equal.

• Conic Sections

  (dx + ey)^{2} + 2gx + 2fy + c = 0

  The general form of the equation of a parabola, in which the second-degree terms form a perfect square.

• This equation is in Coordinate Geometry (Coordinate Geometry)

Problems

No exercises in this chapter.