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Class 11 Math Conic Sections Notes
Chapter 9: Straight Lines Notes
Chapter 11: 3D Geometry Notes
Key Points
Conic Sections
Consider a double napped right circular cone, having semi vertical angle α.
Let β be the angle between the plane and the axis of cone.
When the plane cuts the nappe (other than the vertex) of the cone in different positions, different sections so obtained are called conic sections.
- When β = 90°, section is circle
- When α < β < 90°, section is an ellipse
- When β = α, section is parabola
- When 0 ≤ β < α, section is hyperbola
Degenerated conic sections
When the plane cuts at the vertex of the cone, we have the following different cases:
- When α < β ≤ 90°, section is point
- When β = α, section is straight line
- When 0 ≤ β < α, section is pair of intersecting straight lines
Circle
A circle is the set of all points in a plane that are equidistant from a fixed point in the plane.
- The fixed point is called the centre of circle.
The distance from centre to any point on the circle is called radius of the circle. - The equation of a circle with radius r having centre (h, k) is given by
The general equation of the circle is given by
where g, f and c are constants.
Centre of circle = (-g, -f)
Radius of circle = √(g² + f² − c)
Parabola
A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distance from a fixed line in the plane.
- The fixed point F is called Focus.
The fixed line l is called directrix of the parabola. - A line through the focus and perpendicular to the directrix is called axis of parabola.
- The point of intersection of parabola with the axis is called vertex of parabola.
The distance of any point on the parabola from the focus is called focal length. - Latus rectum – It is a line segment perpendicular to the axis of the parabola, through the focus and whose end points lie on the parabola. The length of the latus rectum = 2 × perpendicular distance of focus from the directrix.
Ellipse
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant.
- The two fixed points are called the foci of the ellipse.
- The mid point of the line segment joining the foci is called the centre of ellipse.
The line segment through the foci of the ellipse is called the major axis. - The line segment through the centre and perpendicular to the major axis is called the minor axis.
- The end points of the major axis are called vertices of the ellipse.
- Eccentricity – It is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse.