Here, you will learn general equation and formulas for conic sections and formula to distinguish between conic.<\/p>\n
A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.<\/p>\n
(a) The fixed point is called the focus.<\/p>\n
(b) The fixed straight line is called directrix.<\/p>\n
(c) The constant ratio is called the eccentricity denoted by e.<\/p>\n
(d) The line passing through the focus & perpendicular to the directrix is called the axis.<\/p>\n
(e) A point of intersection of a conic with its axis is called vertex.<\/p>\n
The general equation of a conic with focus (p,q) & directrix lx + my + n = 0 is :<\/p>\n
\n\\(ax^2 + 2hxy + by^2 + 2gx + 2fy + c\\) = 0<\/p>\n<\/blockquote>\n
Distinguishing between the conic<\/strong><\/p>\n
The nature of conic section depends upon the position of the focus S w.r.t the directrix & also upon the value of eccentricity e. Two different cases arise.<\/p>\n
Case(i) When the focus lies on the directrix :<\/strong><\/p>\n
In this case D = \\(abc + 2fgh – af^2 – bg^2 – ch^2\\) = 0 & the general equation of a conic represent a pair of straight lines and if :<\/p>\n
e > 1, the lines will be real and distinct intersecting at S.<\/p>\n
e = 1, the lines will be coincident.<\/p>\n
e < 1, the lines will be imaginary.<\/p>\n
Case(ii) When the focus does not lie on the directrix :<\/strong><\/p>\n\n\n
The conic represents : \n\t\t\t <\/p>
\n\t\t\t
\n\t\t\t a parabola<\/th>\n\t\t\t an ellipse<\/th> \n\t\t\t a hyperbola<\/th>\n\t\t\t a rectangular hyperbola<\/th>\n\t\t\t <\/tr>\n\t\t\t \n\t\t\t e = 1 ; D \\(\\ne\\) 0<\/td>\n\t\t\t 0 < e < 1 ; D \\(\\ne\\) 0<\/td> \n\t\t\t e > 1 ; D \\(\\ne\\) 0<\/td>\n\t\t\t e > 1 ; D \\(\\ne\\) 0<\/td>\n\t\t\t <\/tr>\n\t\t\t \n\t\t\t \\(h^2\\) = ab<\/td>\n\t\t\t \\(h^2\\) < ab<\/td>\n\t\t\t \\(h^2\\) > ab<\/td>\n\t\t\t \\(h^2\\) > ab ; a + b =0<\/td>\n\t\t\t <\/tr>\n\t\t\t <\/tbody><\/table>\n <\/p>
\n\n\nHope you learnt general equation and formulas for conic sections and formula to distinguish between conic, practice more questions to learn more. Good Luck!<\/p>\n\n\n