Section of a Cone
Let `beta` be the angle made by the intersecting plane with vertical axis of the cone
When the plane cuts the nappe (other than the vertex) of the cone, we have following situations:
(i) When `beta = 90°`, the section is a circle.
(ii) When `alpha < beta < 90°`, the section is an ellipse.
(iii) When `beta = alpha`, the section is a parabola.
(iv) When `0 <= beta < alpha`, the plane cut through both the nappes and intersection is a hyperbola.
Conic Section as Locus :
A conic is the locus of point which moves in a plane such that the ratio of its distance from fixed point focus and fixed line directrix is constant. This constant ratio is called eccentricity (e).
According to different values ofe, conic section is named as
(i) If e = 1, the conic is called a parabola.
(ii) If e < 1, the conic is called an ellipse.
(iii) If e > 1, the conic is called a hyperbola.
(iv) If e = 0, the conic is called a circle.
(v) If `e = oo`, the conic is called a pair of straight lines.
Some Important Terms :
(i) Focus The fixed point is called the focus of the conic section or conic and denoted by S.
(ii) Directrix The fixed straight line is called directrix of the conic.
(iii) Axis The straight line perpendicular to the directrix and passing through the focus is said to be the axis.
(iv) Vertex The point of intersection of the conic section and the axis is called vertex of the conic.
(v) Focal chord Any chord passing through the focus is called focal chord.
(vi) Double ordinate A straight line drawn perpendicular to the axis and terminated at both ends by the curve is a double ordinate.
(vii) Latusrectum The double ordinate passing through the focus is called the latusrectum.
When the plane cuts the nappe (other than the vertex) of the cone, we have following situations:
(i) When `beta = 90°`, the section is a circle.
(ii) When `alpha < beta < 90°`, the section is an ellipse.
(iii) When `beta = alpha`, the section is a parabola.
(iv) When `0 <= beta < alpha`, the plane cut through both the nappes and intersection is a hyperbola.
Conic Section as Locus :
A conic is the locus of point which moves in a plane such that the ratio of its distance from fixed point focus and fixed line directrix is constant. This constant ratio is called eccentricity (e).
According to different values ofe, conic section is named as
(i) If e = 1, the conic is called a parabola.
(ii) If e < 1, the conic is called an ellipse.
(iii) If e > 1, the conic is called a hyperbola.
(iv) If e = 0, the conic is called a circle.
(v) If `e = oo`, the conic is called a pair of straight lines.
Some Important Terms :
(i) Focus The fixed point is called the focus of the conic section or conic and denoted by S.
(ii) Directrix The fixed straight line is called directrix of the conic.
(iii) Axis The straight line perpendicular to the directrix and passing through the focus is said to be the axis.
(iv) Vertex The point of intersection of the conic section and the axis is called vertex of the conic.
(v) Focal chord Any chord passing through the focus is called focal chord.
(vi) Double ordinate A straight line drawn perpendicular to the axis and terminated at both ends by the curve is a double ordinate.
(vii) Latusrectum The double ordinate passing through the focus is called the latusrectum.
Equation of Conic Section
If the focus is `(alpha, beta )`, directrix is ax+ by+ c = 0 and P is any point. Then, ratio of distance of point P from the fixed M point (focus) to fixed line (directrix) is
`(PS)/(PM) = e`
`=> sqrt ((x -alpha)^2 + ( y - beta )^2 ) = e ( ax +by +c)/sqrt(a^2 +b^2 )`
`=> (x -alpha)^2 + ( y -beta)^2 = (e^2(ax +by +c ) ^2)/(a^2 +b^2)`
On simplifying above equation, we get second degree equation
`ax^2 + by^2 + 2gx + 2fy + 2hxy + c = 0`
Here discriminant,
`Delta = abc + 2 fgh -af^2 -bg^2 -ch^2`
Equations will represent different conics, they are given
Case II When `Delta != 0`
Case I When `Delta = 0`
| Condition | Nature of conic |
|---|---|
| `ab- h^2 =0` | A pair of coincident lines |
| `ab- h^2 < 0` | Real and distinct pair of straight lines |
| `ab- h^2 > 0` | Lines are imaginary |
| Condition | Nature of conic |
|---|---|
| h =0, a= b, e=0 | A circle |
| `ab -h^2 =0 , e =1` | A parabola |
| `ab - h^2 > 0 , e < 1` | An ellipse |
| `ab - h^2 < 0 , e > 1` | A hyperbola |
| `ab -h^2 < 0 , a +b = 0 , e = sqrt2` | A rectangular hyperbola |
