Here, you will learn general equation and formulas for conic sections and formula to distinguish between conic.
Conic Sections
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.
(a) The fixed point is called the focus.
(b) The fixed straight line is called directrix.
(c) The constant ratio is called the eccentricity denoted by e.
(d) The line passing through the focus & perpendicular to the directrix is called the axis.
(e) A point of intersection of a conic with its axis is called vertex.
General equation of a conic : Focal directrix property
The general equation of a conic with focus (p,q) & directrix lx + my + n = 0 is :
\(ax^2 + 2hxy + by^2 + 2gx + 2fy + c\) = 0
Distinguishing between the conic
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.
Case(i) When the focus lies on the directrix :
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 :
e > 1, the lines will be real and distinct intersecting at S.
e = 1, the lines will be coincident.
e < 1, the lines will be imaginary.
Case(ii) When the focus does not lie on the directrix :
The conic represents :
| a parabola | an ellipse | a hyperbola | a rectangular hyperbola |
|---|---|---|---|
| e = 1 ; D \(\ne\) 0 | 0 < e < 1 ; D \(\ne\) 0 | e > 1 ; D \(\ne\) 0 | e > 1 ; D \(\ne\) 0 |
| \(h^2\) = ab | \(h^2\) < ab | \(h^2\) > ab | \(h^2\) > ab ; a + b =0 |
Hope you learnt general equation and formulas for conic sections and formula to distinguish between conic, practice more questions to learn more. Good Luck!