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When a plane intersects the nappe of a cone at any place other than the vertex such that the angle b between the plane and the axis of the cone is greater than angle a between the axis and the generator of the cone, but less than 90° the resulting conic section is an ellipse.

Consider two fixed points F1 and F2 in a plane. Now consider points P1, P2 and P3 such that the sum of their distances from F1 and F2 is equal.

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2}

Then the closed curve passing through all such points is an ellipse. Thus, an ellipse is the set of all those points in a plane, the sum of whose distances from two fixed points is constant.

An ellipse is formed only by those points whose sum of distances from two fixed points is constant, and greater than the distance between the two fixed points.

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2} = k (constant) such that k > F_{1}F_{2.}

The fixed points in the plane are the two focus points of an ellipse, jointly called the foci.

The line segment passing through the foci of an ellipse and touching it is called the major axis of the ellipse.

The two points where the major axis touches the ellipse are called the vertices of the ellipse.

The mid-point of the major axis, or the midpoint of the line segment joining the foci of an ellipse, is called the centre of the ellipse.

The line segment passing through the centre of an ellipse and perpendicular to the major axis is called the minor axis of the ellipse.

Semi-major axis = ½ major axis

Semi-minor axis = ½ minor axis

Relationship Between lengths of Major and Minor axes

If major axis AB = 2*a*

Semi-major axes OA = OB = *a*

If minor axis CD = 2*b*

Semi-minor axes OC = OD = *b*

If F_{1}F_{2} = 2*c*

Distance of a focus from centre = OF_{1} = OF_{2} = *c*

Relationship between length of the semi-major axis, length of the semi-minor axis and distance of a focus of an ellipse from its centre:

Consider a point P at one of the end-points of the major axis of an ellipse.

From the figure, PF_{1} = F_{1}O + OP

⇒ PF_{1} = *c* + *a*

From the figure, PF_{2} = OP - OF_{2}

⇒ PF_{2} = *a* - *c*

⇒ PF_{1} + PF_{2} = *c* + *a* + *a* - *c*

⇒ PF_{1} + PF_{2} = 2*a* ------ (1)

Now consider a point Q at one of the end-points of the minor axis of the ellipse.

The distance of point Q from foci F1 and F2 is given by QF1 and QF2.

In right ∆QOF_{1}: QF_{1}^{2} = QO^{2} + OF_{1}^{2}

⇒ QF_{1}^{2} = *b*^{2} + *c*^{2}

⇒ QF_{1} = √(*b*^{2} + *c*^{2})

In right ∆QOF_{2}: QF_{2}^{2} = QO^{2} + OF_{2}^{2}

⇒ QF_{2}^{2} = *b*^{2} + *c*^{2}

⇒ QF_{2} = √(*b*^{2} + *c*^{2})

⇒ QF_{1} + QF_{2} = 2√(*b*^{2} + *c*^{2}) ---- (2)

By definition, PF_{1} + PF_{2} = QF_{1} + QF_{2}

From equations (1) and (2),

⇒ 2a = 2√(b^{2} + c^{2})

⇒ a^{2} = b^{2} + c^{2}

⇒ c^{2} = a^{2} - b^{2 }

⇒ c = √(a^{2} - b^{2})^{ } ----- (3)

c = √(a^{2} - b^{2})

This equation gives an important relation between distance C of a focus of an ellipse from its centre, and its semi-major axis A and semi-minor axis B and helps understand some special cases of an ellipse.

Special Case I: c = 0

If c = 0

⇒ 0 = √(a^{2} - b^{2})^{ }

⇒ a^{2} - b^{2} = 0

⇒ a = b

Thus, if C is equal to zero, the lengths of the semi-major axis and the semi-minor axis of an ellipse become equal, and the ellipse becomes a circle with its foci as the centre.

Special Case II: *c* = *a*

If *c* = *a*

⇒*a* = √(*a*^{2} - *b*^{2})

⇒ *a*^{2} = *a*^{2} - *b*^{2}

⇒ *b* = 0

Thus, if C is equal to A, the length of the semi-minor axis of an ellipse becomes zero, and the ellipse becomes a straight line.

When a plane intersects the nappe of a cone at any place other than the vertex such that the angle b between the plane and the axis of the cone is greater than angle a between the axis and the generator of the cone, but less than 90° the resulting conic section is an ellipse.

Consider two fixed points F1 and F2 in a plane. Now consider points P1, P2 and P3 such that the sum of their distances from F1 and F2 is equal.

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2}

Then the closed curve passing through all such points is an ellipse. Thus, an ellipse is the set of all those points in a plane, the sum of whose distances from two fixed points is constant.

An ellipse is formed only by those points whose sum of distances from two fixed points is constant, and greater than the distance between the two fixed points.

P_{1}F_{1} + P_{1}F_{2} = P_{2}F_{1} + P_{2}F_{2} = P_{3}F_{1} + P_{3}F_{2} = k (constant) such that k > F_{1}F_{2.}

The fixed points in the plane are the two focus points of an ellipse, jointly called the foci.

The line segment passing through the foci of an ellipse and touching it is called the major axis of the ellipse.

The two points where the major axis touches the ellipse are called the vertices of the ellipse.

The mid-point of the major axis, or the midpoint of the line segment joining the foci of an ellipse, is called the centre of the ellipse.

The line segment passing through the centre of an ellipse and perpendicular to the major axis is called the minor axis of the ellipse.

Semi-major axis = ½ major axis

Semi-minor axis = ½ minor axis

Relationship Between lengths of Major and Minor axes

If major axis AB = 2*a*

Semi-major axes OA = OB = *a*

If minor axis CD = 2*b*

Semi-minor axes OC = OD = *b*

If F_{1}F_{2} = 2*c*

Distance of a focus from centre = OF_{1} = OF_{2} = *c*

Relationship between length of the semi-major axis, length of the semi-minor axis and distance of a focus of an ellipse from its centre:

Consider a point P at one of the end-points of the major axis of an ellipse.

From the figure, PF_{1} = F_{1}O + OP

⇒ PF_{1} = *c* + *a*

From the figure, PF_{2} = OP - OF_{2}

⇒ PF_{2} = *a* - *c*

⇒ PF_{1} + PF_{2} = *c* + *a* + *a* - *c*

⇒ PF_{1} + PF_{2} = 2*a* ------ (1)

Now consider a point Q at one of the end-points of the minor axis of the ellipse.

The distance of point Q from foci F1 and F2 is given by QF1 and QF2.

In right ∆QOF_{1}: QF_{1}^{2} = QO^{2} + OF_{1}^{2}

⇒ QF_{1}^{2} = *b*^{2} + *c*^{2}

⇒ QF_{1} = √(*b*^{2} + *c*^{2})

In right ∆QOF_{2}: QF_{2}^{2} = QO^{2} + OF_{2}^{2}

⇒ QF_{2}^{2} = *b*^{2} + *c*^{2}

⇒ QF_{2} = √(*b*^{2} + *c*^{2})

⇒ QF_{1} + QF_{2} = 2√(*b*^{2} + *c*^{2}) ---- (2)

By definition, PF_{1} + PF_{2} = QF_{1} + QF_{2}

From equations (1) and (2),

⇒ 2a = 2√(b^{2} + c^{2})

⇒ a^{2} = b^{2} + c^{2}

⇒ c^{2} = a^{2} - b^{2 }

⇒ c = √(a^{2} - b^{2})^{ } ----- (3)

c = √(a^{2} - b^{2})

This equation gives an important relation between distance C of a focus of an ellipse from its centre, and its semi-major axis A and semi-minor axis B and helps understand some special cases of an ellipse.

Special Case I: c = 0

If c = 0

⇒ 0 = √(a^{2} - b^{2})^{ }

⇒ a^{2} - b^{2} = 0

⇒ a = b

Thus, if C is equal to zero, the lengths of the semi-major axis and the semi-minor axis of an ellipse become equal, and the ellipse becomes a circle with its foci as the centre.

Special Case II: *c* = *a*

If *c* = *a*

⇒*a* = √(*a*^{2} - *b*^{2})

⇒ *a*^{2} = *a*^{2} - *b*^{2}

⇒ *b* = 0

Thus, if C is equal to A, the length of the semi-minor axis of an ellipse becomes zero, and the ellipse becomes a straight line.