2

As per book every conic have 4 foci ,two real and 2 imaginary. I cannot understand and visualize this.

soniaverma
  • 31
  • Imaginary foci arise when considering conics on the "complex projective plane". If it's any consolation, I have trouble understanding and visualizing stuff there, myself. :) Conics in the real plane have two foci; they coincide on the circle, and one of them is out at infinity for the parabola. – Blue Oct 04 '18 at 12:06
  • I came across a short (three-page) and possibly-interesting journal article: "Imaginary Focal Properties of Conics" (JSTOR). It may or may not help you (or me!) visualize things. – Blue Oct 04 '18 at 12:14
  • I have some more questions now. What's the significance of foci in real plane. Do the incident rays converge at these points after reflection. If yes how can we prove it geometrically or algebraically? – soniaverma Oct 04 '18 at 12:20
  • If you search Math.SE for "reflection property", you'll find a number of previously-posted questions on the topic. See if the answers to those provide the insights you need. – Blue Oct 04 '18 at 12:38
  • Thanks a lot for the help. In the article mentioned above the explanation started by keeping the source of light at the focus. My question is about proving the existence of focus. – soniaverma Oct 04 '18 at 13:03
  • To be more specific,if I give a quadratic polynomial , say, x^2 -5x +6 to plot , variable (along x axis) VS value of polynomial(along y axis), then it will result in parabola.Is there any geometric way out to find the point of convergence of rays parallel to axis of symmetry. – – soniaverma Oct 04 '18 at 13:05
  • After hitting a parabola a ray parallel to axis is reflected so that angles of incidence and reflection are equal by the reflection law. Focus point is intersection point of any two reflected rays.Geometric construction can based on this fact. Next, unlike a parabola we have not a single point focus for hyperbola and ellipse.. the off-axis rays do not converge well and there is a caustic envelope. – Narasimham Oct 04 '18 at 20:07
  • You might find the Wikipedia article on parabolas a useful starting point. – amd Oct 05 '18 at 00:23

0 Answers0