Hi All
AoA
I need help on Derivation of equation of ellips and hyperbola.
Any article, site regarding my question plz??
Thanx
Usman
Since this is not my area of expertise, I searched Google under the key words "ellipse hyperbola equations" to get these possible sources:
http://en.wikipedia.org/wiki/Ellipse
(Broken Link Removed)
http://www.schillerinstitute.org/fid_02-06/2006/061-2_375_Kepler.html
http://www.answers.com/topic/ellipse
I hope it helps. Thanks for asking.
how do we get equation of ellips?
Hello Usman,
I understand that you are looking for help with the derivation of equations of ellipses and hyperbolas. While I cannot provide you with a direct article or website link, I can explain the general process of deriving these equations.
The equation of an ellipse is derived from the definition of an ellipse as the set of all points in a plane, the sum of whose distances to two fixed points (called foci) is constant. Let's consider an ellipse centered at the origin with semi-major axis 'a' and semi-minor axis 'b'. The foci lie along the x-axis at (-c, 0) and (c, 0), where c = √(a^2 - b^2).
To derive the equation of the ellipse, you can consider a general point on the ellipse as (x, y). Using the distance formula, the sum of the distances from (x, y) to the foci is equal to the constant distance 2a, which is the length of the major axis. This gives us:
√((x + c)^2 + y^2) + √((x - c)^2 + y^2) = 2a
Squaring both sides and simplifying the equation will result in the standard equation of an ellipse:
(x^2 / a^2) + (y^2 / b^2) = 1
For a hyperbola, the equation is derived from the definition of a hyperbola as the set of all points in a plane, the difference of whose distances to two fixed points (called foci) is constant. Let's consider a hyperbola centered at the origin with semi-transverse axis 'a' and semi-conjugate axis 'b'. The foci lie along the x-axis at (-c, 0) and (c, 0), where c = √(a^2 + b^2).
Using a similar approach as with the ellipse, you can consider a general point on the hyperbola as (x, y) and use the distance formula to derive the equation. The resulting standard equation of a hyperbola is:
(x^2 / a^2) - (y^2 / b^2) = 1
I hope this explanation helps you understand the derivation of the equations of ellipses and hyperbolas. Please let me know if you have any further questions!