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When a plane intersects a cone at any place other than the vertex, such that the angle b between the plane and the axis of the cone is less than the angle a between the axis and the generator of the cone, and more than or equal to zero degrees, and the plane intersects both the nappes of the cone, the resulting conic section is a hyperbola.

A hyperbola is made of two curves that appear like mirror images of each other.

Consider two fixed points F1 and F2 in a plane.

Now consider points P1, P2 and P3 such that the difference of their distances from F1 and F2 is equal. Then the curves passing through all such points form a hyperbola.

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

A hyperbola is the set of all the points in a plane the difference of whose distances from two fixed points in the plane is constant.

To find the difference between the distances of a point from the fixed points F1 and F2, we always subtract the distance of the nearer fixed point from the distance of the farther fixed point.

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

The fixed points in the plane are the two focus of the hyperbola. They are jointly called the foci of the hyperbola.

The line passing through the foci is called the transverse axis of the hyperbola.

The two points where the transverse axis intersects the curves of a hyperbola are called the vertices of the hyperbola.

The mid-point of the line segment joining the foci is called the centre of the hyperbola.

The line passing through the centre and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.

Relationship between Transverse and Conjugate Axes

The distance between the two vertices of a hyperbola is considered the length of its transverse axis.

If AB = 2*a*

OA = OB = *a*

Length of semi-transverse axis = ½ Length of transverse axis

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

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

An asymptote is a tangent to the hyperbola at infinity.

If a tangent is drawn to the hyperbola through any of its vertices, the tangent intersects the asymptotes as shown.

Perpendiculars are drawn on the conjugate axis from the points where the tangent intersects the asymptotes.

The distance between the two points where these perpendiculars meet the conjugate axis is considered the length of the conjugate axis of the hyperbola.

The length of the conjugate axis of a hyperbola is represented by 2b, and gets bisected by the centre.

If CD = 2*b*

OC = OD = *b*

The centre bisects the conjugate axis into two semi-conjugate axes.

Length of semi-conjugate axis = ½ Length of conjugate axis.

Relationship between lengths A, B and C:

If the lengths of the semi-transverse and semi-conjugate axes of a hyperbola are equal to A and B, respectively, and the distance between its foci is 2C, then

*b*^{2} = *c*^{2} - *a*^{2 }

or *b* = √(*c*^{2} - *a*^{2})^{ }

Difference of the distances of a point on a hyperbola from its foci = P_{1}F_{2} - P_{1}F_{1} = P_{2}F_{2} - P_{2}F_{1} = P_{3}F_{1} - P_{3}F_{2} = k.

By definition, the difference between the distances of a point on a hyperbola from its foci is constant. Let this constant be k.

Consider vertex A that lies on the hyperbola.

Difference of distances of point A from F_{1} and F_{2} =k = AF_{2} - AF_{1}

⇒ k = AB + BF_{2} - AF_{1} ……(1)

Now consider the other vertex B that lies on the hyperbola.

Difference of distances of point B from F_{1} and F_{2}=k = BF_{1} - BF_{2}

⇒ k = AB + AF_{1} - BF_{2} ……(2)

From equations (1) and (2), we get

AB + BF_{2} - AF_{1} = AB + AF_{1} - BF_{2}

⇒ BF_{2} - AF_{1} = AF_{1} - BF_{2}

⇒ BF_{2} = AF_{1}

From equation (1),

k = AB + BF_{2} - AF_{1}

⇒ k = 2*a* + AF_{1} - AF_{1}

⇒ k = 2*a*

From equation (2),

k = AB + AF_{1} - BF_{2}

⇒ k = 2*a* + AF_{1} - AF_{1}

⇒ k = 2*a*

PF_{1} - PF_{2} = 2*a*

This is true for all the points on a hyperbola.

The difference between the distances of any point on a hyperbola from its foci is equal to the length of the transverse axis of the hyperbola.

When a plane intersects a cone at any place other than the vertex, such that the angle b between the plane and the axis of the cone is less than the angle a between the axis and the generator of the cone, and more than or equal to zero degrees, and the plane intersects both the nappes of the cone, the resulting conic section is a hyperbola.

A hyperbola is made of two curves that appear like mirror images of each other.

Consider two fixed points F1 and F2 in a plane.

Now consider points P1, P2 and P3 such that the difference of their distances from F1 and F2 is equal. Then the curves passing through all such points form a hyperbola.

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

A hyperbola is the set of all the points in a plane the difference of whose distances from two fixed points in the plane is constant.

To find the difference between the distances of a point from the fixed points F1 and F2, we always subtract the distance of the nearer fixed point from the distance of the farther fixed point.

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

The fixed points in the plane are the two focus of the hyperbola. They are jointly called the foci of the hyperbola.

The line passing through the foci is called the transverse axis of the hyperbola.

The two points where the transverse axis intersects the curves of a hyperbola are called the vertices of the hyperbola.

The mid-point of the line segment joining the foci is called the centre of the hyperbola.

The line passing through the centre and perpendicular to the transverse axis is called the conjugate axis of the hyperbola.

Relationship between Transverse and Conjugate Axes

The distance between the two vertices of a hyperbola is considered the length of its transverse axis.

If AB = 2*a*

OA = OB = *a*

Length of semi-transverse axis = ½ Length of transverse axis

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

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

An asymptote is a tangent to the hyperbola at infinity.

If a tangent is drawn to the hyperbola through any of its vertices, the tangent intersects the asymptotes as shown.

Perpendiculars are drawn on the conjugate axis from the points where the tangent intersects the asymptotes.

The distance between the two points where these perpendiculars meet the conjugate axis is considered the length of the conjugate axis of the hyperbola.

The length of the conjugate axis of a hyperbola is represented by 2b, and gets bisected by the centre.

If CD = 2*b*

OC = OD = *b*

The centre bisects the conjugate axis into two semi-conjugate axes.

Length of semi-conjugate axis = ½ Length of conjugate axis.

Relationship between lengths A, B and C:

If the lengths of the semi-transverse and semi-conjugate axes of a hyperbola are equal to A and B, respectively, and the distance between its foci is 2C, then

*b*^{2} = *c*^{2} - *a*^{2 }

or *b* = √(*c*^{2} - *a*^{2})^{ }

Difference of the distances of a point on a hyperbola from its foci = P_{1}F_{2} - P_{1}F_{1} = P_{2}F_{2} - P_{2}F_{1} = P_{3}F_{1} - P_{3}F_{2} = k.

By definition, the difference between the distances of a point on a hyperbola from its foci is constant. Let this constant be k.

Consider vertex A that lies on the hyperbola.

Difference of distances of point A from F_{1} and F_{2} =k = AF_{2} - AF_{1}

⇒ k = AB + BF_{2} - AF_{1} ……(1)

Now consider the other vertex B that lies on the hyperbola.

Difference of distances of point B from F_{1} and F_{2}=k = BF_{1} - BF_{2}

⇒ k = AB + AF_{1} - BF_{2} ……(2)

From equations (1) and (2), we get

AB + BF_{2} - AF_{1} = AB + AF_{1} - BF_{2}

⇒ BF_{2} - AF_{1} = AF_{1} - BF_{2}

⇒ BF_{2} = AF_{1}

From equation (1),

k = AB + BF_{2} - AF_{1}

⇒ k = 2*a* + AF_{1} - AF_{1}

⇒ k = 2*a*

From equation (2),

k = AB + AF_{1} - BF_{2}

⇒ k = 2*a* + AF_{1} - AF_{1}

⇒ k = 2*a*

PF_{1} - PF_{2} = 2*a*

This is true for all the points on a hyperbola.

The difference between the distances of any point on a hyperbola from its foci is equal to the length of the transverse axis of the hyperbola.