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Chapter – 8
Conic Section
Let O be the fixed point and OC, the fixed line. Then the surface generated by rotating the line OA around OC such that ∠AOC is always constant, is known as the right cone. The point O is known as the vertex, OC, the axis and OA, the generator. ZAOC is known as the semi- vertical angle.
Conic Section
If the cone OA’B’ is symmetrical to the cone OAB about OC’ opposite to OC, then ABOA’B’ is said to be the double right cone.
Conic Section Class 12 Mathematics Notes PDF
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The closed or the open curve obtained by the intersection of the cone and the plane is the conic section. If a plane cuts the cone, a curve will be obtained. The nature of the curve depends upon the position of the cutting plane. The following are the curves (conic section) obtained when a cone is intersected by a plane in different positions.
- If a plane intersects a cone perpendicular to the axis, then the section is a circle.
- If a plane intersects a cone at a given angle with the axis greater than the semi-vertical angle, then the section is an ellipse.
- If an intersecting plane, not passing through the vertex, is parallel to the generator of the cone, then the section is a parabola.
- If a plane intersects the double right cone such that the angle between the axis and the plane be less than the semi-vertical angle, then the section is a hyperbola.
Usually, a conic section is defined in the following ways: The locus of a point which moves in a plane in such a way that the ratio of its distance.
The fixed point is called the focus, the fixed straight line its directrix, and the constant ratio the eccentricity (denoted be e). The straight line passing through the focus and perpendicular to the directrix is called the axis. The intersection of the curve and the axis is called the vertex.
Circle
The locus determined by the moving point such that its distance from the fixed point is always constant is known as the circle. The fixed point is known as the centre and the constant distance, the radius of the circle. That is if O is the fixd point and P, any point one the locus such that OP is constant then OP is the radius of the circle.
A Line and a Circle
Let y = mx + c and x² + y² = a² be the equations of a line and a circle respectively. If the line intersects the circle, the points of intersection can be obtained by solving the two equations simultaneously. Thus, using y = mx + c in x² + y² = a² , we get x² + (mx + c)² = a²
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