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Circle & Conics

Circle and its Equation in different forms

• Conic sections or conics are the curves obtained by intersecting a double-napped right-circular cone with a plane.

• The concept of conic sections is widely used in astronomy, projectile motion of an object, etc.

• The example of conic sections are circle (Figure I), ellipse (Figure II), parabola (Figure III) and hyperbola (Figure IV). • Different types of conics can be formed by intersecting a plane with a double-napped cone (other than the vertex) by different ways.

• If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.

 Condition Conic Formed Figure θ2 = 90° (The plane cuts only one nappe of the cone entirely) Circle θ1 < θ2 < 90° (The plane cuts only one nappe of the cone entirely) Ellipse θ1 = θ2 (The plane cuts only one nappe of the cone entirely) Parabola 0 ≤ θ2 < θ1 (The plane cuts each nappe of the cone entirely) Hyperbola • The conic sections obtained by cutting a plane with a double-napped cone at its vertex are known as degenerated conic sections.

• If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.

 Condition Conic Formed Figure θ1 < θ2 ≤ 90° Point θ1 =θ2 Line 0 ≤ θ2 < θ1 Hyperbola • Conic sections or conics are the curves obtained by intersecting a double-napped right-circular cone with a plane.

• The concept of conic sections is widely used in astronomy, projectile motion of an object, etc.

• The example of conic sections are circle (Figure I), ellipse (Figure II), parabola (Figure III) and hyperbola (Figure IV). • Different types of conics can be formed by intersecting a plane with a double-napped cone (other than the vertex) by different ways.

• If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.

 Condition Conic Formed Figure θ2 = 90° (The plane cuts only one nappe of the cone entirely) Circle θ1 < θ2 < 90° (The plane cuts only one nappe of the cone entirely) Ellipse θ1 = θ2 (The plane cuts only one nappe of the cone entirely) Parabola 0 ≤ θ2 < θ1 (The plane cuts each nappe of the cone entirely) Hyperbola • The conic sections obtained by cutting a plane with a double-napped cone at its vertex are known as degenerated conic sections.

• If θ1 is the angle between the axis and the generator and θ2 is the angle between the plane and the axis, then for different conditions of θ1 and θ2, we get different conics. These are described in the table shown below.

… Condition Conic Formed Figure θ1 < θ2 ≤ 90° Point θ1 =θ2 Line To view the complete topic, please

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