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A hyperbola is the set of all that points in a plane such that the difference of their distances from two fixed points in the plane is constant.** **

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

Foci, vertices, centre, transverse axis and conjugate axis of a hyperbola are as shown in the figure.

In a hyperbola, *b*^{2} = *c*^{2} - *a*^{2},^{ }

Where

*c* = Distance of a focus from the centre

*a* = Length of semi-transverse axis

*b* = Length of semi-conjugate axis

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.

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

Eccentricity is a measure of the deviation of a conic section from being a circle.

The eccentricity of a hyperbola is denoted by E, and is equal to the ratio of the distance of a focus from its centre and the length of its semi-transverse axis.

Eccentricity of a hyperbola (*e*) = *c*/*a*

Þ c = ea

Thus, the distance of a focus from the centre of a hyperbola is equal to the product of its eccentricity and the length of its semi-transverse axis.

**Standard Equations of a Hyperbola**

Consider a hyperbola with its centre at the origin and its transverse axis along the X- or the Y-axis, and then there are two distinct possibilities.

In the first case, we can have a hyperbola with its centre lying on the origin and its transverse axis along the X-axis. In this case, the foci of the hyperbola lie on the X-axis.

In the second case, we can have a hyperbola with its centre lying on the origin and its transverse axis along the Y-axis. In this case, the foci of the hyperbola lie on the Y-axis.

Case (i)

**Equation of the hyperbola:**

Consider a point P on the hyperbola with the coordinates X, Y.

Then, by definition, the difference of the distances PF1 and PF2 is constant and equal to 2A.

PF_{1} - PF_{2} = 2*a* ……(1)

The distance between the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) = √(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2}

Using distance formula, PF_{1} = √{(*x* - (-*c*)}^{2} + (*y* - 0)^{2}

⇒ PF_{1} = √(*x* + *c*)^{2} + *y*^{2} ……(2)

Using distance formula, PF_{2} = √(*x* - *c*)}^{2} + (*y* - 0)^{2}

⇒ PF_{2} = √(*x* - *c*)^{2} + *y*^{2} ……(3)

From equations (1), (2) and (3), we get

√(*x* + *c*)^{2} + *y*^{2} - √(*x* - *c*)^{2} + *y*^{2} = 2*a*

⇒ √(*x* + *c*)^{2} + *y*^{2} = 2*a* + √(*x* - *c*)^{2} + *y*^{2}

Squaring both sides,

(*x* + *c*)^{2} + *y*^{2} = 4*a*^{2} + 4*a*√(*x* - *c*)^{2} + *y*^{2} + (*x* - *c*)^{2} + *y*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = (*x* + *c*)^{2} - (*x* - *c*)^{2} - 4*a*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = *x*^{2} + *c*^{2} + 2*xc* - *x*^{2} - *c*^{2} + 2*xc* - 4*a*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = 4*xc* ^{ }- 4*a*^{2}

⇒ *a*√(*x* - *c*)^{2} + *y*^{2} = *xc* - *a*^{2}

⇒ √(*x* - *c*)^{2} + *y*^{2} = c/a *x* - *a*

Squaring both sides,

(*x* - *c*)^{2} + *y*^{2} = *a*^{2} + c^{2}/a^{2}* x* - 2*cx*

⇒*x*^{2} + *c*^{2} - 2*cx* + *y*^{2} = *a*^{2} + c^{2}/a^{2 }*x*^{2} - 2*cx*

⇒*x*^{2} + *c*^{2} + *y*^{2} = *a*^{2} + c^{2}/a^{2 }*x*^{2}

⇒*x*^{2} - c^{2}/a^{2 }*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒(1 - c^{2}/a^{2})*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒((a^{2} - c^{2})/a^{2}) *x*^{2} + *y*^{2} = *a*^{2} - *c*^{2} ……(4)

⇒(^{ }- b^{2}/a^{2}*) x*^{2} + *y*^{2} = -*b*^{2 } (Since *b*^{2} = *c*^{2} - *a*^{2})

Multiplying both sides by -1/*b*^{2},

x^{2}/a^{2} - y^{2}/b^{2} = 1 ……(5)

Therefore, equation x^{2}/a^{2} - y^{2}/b^{2} = 1 represents the given hyperbola if the coordinates of point P as derived from this equation satisfy the geometrical condition (PF_{1} - PF_{2}) = 2a.

From equation (5),

*y*^{2} = *b*^{2} (x^{2}/a^{2} - 1)

From equation (2),

PF_{1} = √(*x* + *c*)^{2} + *y*^{2}

⇒ PF_{1} = √(*x* + *c*)^{2} + *b*^{2} (x^{2}/a^{2} - 1)

= √ *x*^{2} + *c*^{2} + 2*cx* + *b*^{2}x^{2}/a^{2} - *b*^{2} ^{ }

= √ *x*^{2}(1+ *b*^{2}/a^{2} + 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √ *x*^{2}((a^{2}+*b*^{2})/a^{2})+ 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √*c*^{2}/a^{2} .x^{2}+ 2*cx* + *a*^{2} (Since *b*^{2} = *c*^{2} - *a*^{2})

= √(a + cx/a)^{2}

⇒ PF_{1} = a + cx/a

From equation (3),

PF_{2} = √(*x* - *c*)^{2} + *y*^{2}

⇒ PF_{1} = √(*x* - *c*)^{2} + *b*^{2} (*x*^{2}/a^{2} - 1)

= √ *x*^{2} + *c*^{2} - 2*cx* + *b*^{2}*x*^{2}/a^{2} - *b*^{2} ^{ }

= √ *x*^{2}(1+ *b*^{2}/a^{2}) - 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √ *x*^{2}( (a^{2}+ *b*^{2})/a^{2}) - 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √*c*^{2}/a^{2} . *x*^{2} - 2*cx* + *a*^{2} (Since *b*^{2} = *c*^{2} - *a*^{2})

= √( c/a x - a)^{2}

⇒ PF_{2} = c/a x - *a*

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

**Case I:** The equation, x^{2}/a^{2} - y^{2}/b^{2} = 1, is the standard equation for a hyperbola with its centre at the origin and the transverse axis lying along the X-axis.

**Case II:** Similarly, the standard equation for a hyperbola with its centre at the origin and the transverse axis along the Y-axis is y^{2}/a^{2} - x^{2}/b^{2} = 1.

These two equations are called the standard equations of a hyperbola having its centre at the origin and the transverse axis along the X- or Y-axis.

If the coefficient of *x*^{2} is positive, then the transverse axis lies along the *X*-axis.

If the coefficient of *y*^{2} is positive, then the transverse axis lies along the *Y*-axis.

Note that both the standard equations of a hyperbola contain even powers of x and y.

A hyperbola with its centre lying on the origin and the transverse axis along either the *X-* or the Y-axis is symmetrical about both the coordinate axes.

⇒ If (*x*, *y*) lies on the hyperbola, then (-*x*, *y*), (*x*, -*y*) and (-*x*, -*y*) also lie on it.

If *a* = *b*, then the hyperbola is called an equilateral hyperbola.

**Case I:** Centre (0, 0), transverse axis along the *X*-axis.

x^{2}/a^{2} - y^{2}/b^{2} = 1

⇒ x^{2}/a^{2} = 1 + y^{2}/b^{2}

⇒ |x/a | ≥ 1

⇒x ≤ -*a* or x ≥ *a*

⇒ No point on the curve of the hyperbola lies between the lines *x* = -*a* and *x* = *a*.

**Case II:** Centre (0, 0), transverse axis along the *Y*-axis.

y^{2}/a^{2} - x^{2}/b^{2} = 1

⇒ y^{2}/a^{2} = 1 + x^{2}/b^{2}

⇒ |y/a |≥ 1

⇒ y ≤ -a or y ≥ *a*

⇒ No point on the curve of the hyperbola lies between the lines *y* = -*a* and *y* = *a*.

**Expression for Length of Latus Rectum**

A line segment passing through a focus and perpendicular to the transverse axis, and with its end points lying on the curve of a hyperbola, is called its latus rectum.

Since a hyperbola has two foci, it has two latus recta.

Consider a hyperbola with its centre at the origin and the transverse axis along the X-axis and latus recta AB and CD.

Let AF_{2} = l

⇒ Coordinates of A are (*c*, l), where

*c* = Distance of focus F_{2} from centre O

Equation of the given hyperbola: x^{2}/a^{2} - y^{2}/b^{2} = 1

Sine A (*c*, l) lies on x^{2}/a^{2} - y^{2}/b^{2} = 1

⇒ c^{2}/a^{2} - l^{2}/b^{2} = 1

⇒ l^{2} = b^{2}(c^{2}/a^{2} - 1)

⇒ l^{2} = b^{2}((c^{2}-a^{2}) / a^{2})

⇒ l^{2} = b^{2}( b^{2}/ a^{2}) (Since *b*^{2} = *c*^{2} - *a*^{2})

⇒ l^{2} = b^{4}/ a^{2}

⇒ l = b^{2}/a

AF_{2} = F_{2}B = b^{2}/a

⇒ AB = AF_{2} + F_{2}B = 2b^{2}/a

Length of latus rectum of a hyperbola = 2b^{2}/a,

Where,

*a* = Length of semi-transverse axis

*b* = Length of semi-conjugate axis.

A hyperbola is the set of all that points in a plane such that the difference of their distances from two fixed points in the plane is constant.** **

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

Foci, vertices, centre, transverse axis and conjugate axis of a hyperbola are as shown in the figure.

In a hyperbola, *b*^{2} = *c*^{2} - *a*^{2},^{ }

Where

*c* = Distance of a focus from the centre

*a* = Length of semi-transverse axis

*b* = Length of semi-conjugate axis

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.

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

Eccentricity is a measure of the deviation of a conic section from being a circle.

The eccentricity of a hyperbola is denoted by E, and is equal to the ratio of the distance of a focus from its centre and the length of its semi-transverse axis.

Eccentricity of a hyperbola (*e*) = *c*/*a*

Þ c = ea

Thus, the distance of a focus from the centre of a hyperbola is equal to the product of its eccentricity and the length of its semi-transverse axis.

**Standard Equations of a Hyperbola**

Consider a hyperbola with its centre at the origin and its transverse axis along the X- or the Y-axis, and then there are two distinct possibilities.

In the first case, we can have a hyperbola with its centre lying on the origin and its transverse axis along the X-axis. In this case, the foci of the hyperbola lie on the X-axis.

In the second case, we can have a hyperbola with its centre lying on the origin and its transverse axis along the Y-axis. In this case, the foci of the hyperbola lie on the Y-axis.

Case (i)

**Equation of the hyperbola:**

Consider a point P on the hyperbola with the coordinates X, Y.

Then, by definition, the difference of the distances PF1 and PF2 is constant and equal to 2A.

PF_{1} - PF_{2} = 2*a* ……(1)

The distance between the points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) = √(*x*_{2} - *x*_{1})^{2} + (*y*_{2} - *y*_{1})^{2}

Using distance formula, PF_{1} = √{(*x* - (-*c*)}^{2} + (*y* - 0)^{2}

⇒ PF_{1} = √(*x* + *c*)^{2} + *y*^{2} ……(2)

Using distance formula, PF_{2} = √(*x* - *c*)}^{2} + (*y* - 0)^{2}

⇒ PF_{2} = √(*x* - *c*)^{2} + *y*^{2} ……(3)

From equations (1), (2) and (3), we get

√(*x* + *c*)^{2} + *y*^{2} - √(*x* - *c*)^{2} + *y*^{2} = 2*a*

⇒ √(*x* + *c*)^{2} + *y*^{2} = 2*a* + √(*x* - *c*)^{2} + *y*^{2}

Squaring both sides,

(*x* + *c*)^{2} + *y*^{2} = 4*a*^{2} + 4*a*√(*x* - *c*)^{2} + *y*^{2} + (*x* - *c*)^{2} + *y*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = (*x* + *c*)^{2} - (*x* - *c*)^{2} - 4*a*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = *x*^{2} + *c*^{2} + 2*xc* - *x*^{2} - *c*^{2} + 2*xc* - 4*a*^{2}

⇒ 4*a*√(*x* - *c*)^{2} + *y*^{2} = 4*xc* ^{ }- 4*a*^{2}

⇒ *a*√(*x* - *c*)^{2} + *y*^{2} = *xc* - *a*^{2}

⇒ √(*x* - *c*)^{2} + *y*^{2} = c/a *x* - *a*

Squaring both sides,

(*x* - *c*)^{2} + *y*^{2} = *a*^{2} + c^{2}/a^{2}* x* - 2*cx*

⇒*x*^{2} + *c*^{2} - 2*cx* + *y*^{2} = *a*^{2} + c^{2}/a^{2 }*x*^{2} - 2*cx*

⇒*x*^{2} + *c*^{2} + *y*^{2} = *a*^{2} + c^{2}/a^{2 }*x*^{2}

⇒*x*^{2} - c^{2}/a^{2 }*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒(1 - c^{2}/a^{2})*x*^{2} + *y*^{2} = *a*^{2} - *c*^{2}

⇒((a^{2} - c^{2})/a^{2}) *x*^{2} + *y*^{2} = *a*^{2} - *c*^{2} ……(4)

⇒(^{ }- b^{2}/a^{2}*) x*^{2} + *y*^{2} = -*b*^{2 } (Since *b*^{2} = *c*^{2} - *a*^{2})

Multiplying both sides by -1/*b*^{2},

x^{2}/a^{2} - y^{2}/b^{2} = 1 ……(5)

Therefore, equation x^{2}/a^{2} - y^{2}/b^{2} = 1 represents the given hyperbola if the coordinates of point P as derived from this equation satisfy the geometrical condition (PF_{1} - PF_{2}) = 2a.

From equation (5),

*y*^{2} = *b*^{2} (x^{2}/a^{2} - 1)

From equation (2),

PF_{1} = √(*x* + *c*)^{2} + *y*^{2}

⇒ PF_{1} = √(*x* + *c*)^{2} + *b*^{2} (x^{2}/a^{2} - 1)

= √ *x*^{2} + *c*^{2} + 2*cx* + *b*^{2}x^{2}/a^{2} - *b*^{2} ^{ }

= √ *x*^{2}(1+ *b*^{2}/a^{2} + 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √ *x*^{2}((a^{2}+*b*^{2})/a^{2})+ 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √*c*^{2}/a^{2} .x^{2}+ 2*cx* + *a*^{2} (Since *b*^{2} = *c*^{2} - *a*^{2})

= √(a + cx/a)^{2}

⇒ PF_{1} = a + cx/a

From equation (3),

PF_{2} = √(*x* - *c*)^{2} + *y*^{2}

⇒ PF_{1} = √(*x* - *c*)^{2} + *b*^{2} (*x*^{2}/a^{2} - 1)

= √ *x*^{2} + *c*^{2} - 2*cx* + *b*^{2}*x*^{2}/a^{2} - *b*^{2} ^{ }

= √ *x*^{2}(1+ *b*^{2}/a^{2}) - 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √ *x*^{2}( (a^{2}+ *b*^{2})/a^{2}) - 2*cx* + *c*^{2} - *b*^{2} ^{ }

= √*c*^{2}/a^{2} . *x*^{2} - 2*cx* + *a*^{2} (Since *b*^{2} = *c*^{2} - *a*^{2})

= √( c/a x - a)^{2}

⇒ PF_{2} = c/a x - *a*

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

**Case I:** The equation, x^{2}/a^{2} - y^{2}/b^{2} = 1, is the standard equation for a hyperbola with its centre at the origin and the transverse axis lying along the X-axis.

**Case II:** Similarly, the standard equation for a hyperbola with its centre at the origin and the transverse axis along the Y-axis is y^{2}/a^{2} - x^{2}/b^{2} = 1.

These two equations are called the standard equations of a hyperbola having its centre at the origin and the transverse axis along the X- or Y-axis.

If the coefficient of *x*^{2} is positive, then the transverse axis lies along the *X*-axis.

If the coefficient of *y*^{2} is positive, then the transverse axis lies along the *Y*-axis.

Note that both the standard equations of a hyperbola contain even powers of x and y.

A hyperbola with its centre lying on the origin and the transverse axis along either the *X-* or the Y-axis is symmetrical about both the coordinate axes.

⇒ If (*x*, *y*) lies on the hyperbola, then (-*x*, *y*), (*x*, -*y*) and (-*x*, -*y*) also lie on it.

If *a* = *b*, then the hyperbola is called an equilateral hyperbola.

**Case I:** Centre (0, 0), transverse axis along the *X*-axis.

x^{2}/a^{2} - y^{2}/b^{2} = 1

⇒ x^{2}/a^{2} = 1 + y^{2}/b^{2}

⇒ |x/a | ≥ 1

⇒x ≤ -*a* or x ≥ *a*

⇒ No point on the curve of the hyperbola lies between the lines *x* = -*a* and *x* = *a*.

**Case II:** Centre (0, 0), transverse axis along the *Y*-axis.

y^{2}/a^{2} - x^{2}/b^{2} = 1

⇒ y^{2}/a^{2} = 1 + x^{2}/b^{2}

⇒ |y/a |≥ 1

⇒ y ≤ -a or y ≥ *a*

⇒ No point on the curve of the hyperbola lies between the lines *y* = -*a* and *y* = *a*.

**Expression for Length of Latus Rectum**

A line segment passing through a focus and perpendicular to the transverse axis, and with its end points lying on the curve of a hyperbola, is called its latus rectum.

Since a hyperbola has two foci, it has two latus recta.

Consider a hyperbola with its centre at the origin and the transverse axis along the X-axis and latus recta AB and CD.

Let AF_{2} = l

⇒ Coordinates of A are (*c*, l), where

*c* = Distance of focus F_{2} from centre O

Equation of the given hyperbola: x^{2}/a^{2} - y^{2}/b^{2} = 1

Sine A (*c*, l) lies on x^{2}/a^{2} - y^{2}/b^{2} = 1

⇒ c^{2}/a^{2} - l^{2}/b^{2} = 1

⇒ l^{2} = b^{2}(c^{2}/a^{2} - 1)

⇒ l^{2} = b^{2}((c^{2}-a^{2}) / a^{2})

⇒ l^{2} = b^{2}( b^{2}/ a^{2}) (Since *b*^{2} = *c*^{2} - *a*^{2})

⇒ l^{2} = b^{4}/ a^{2}

⇒ l = b^{2}/a

AF_{2} = F_{2}B = b^{2}/a

⇒ AB = AF_{2} + F_{2}B = 2b^{2}/a

Length of latus rectum of a hyperbola = 2b^{2}/a,

Where,

*a* = Length of semi-transverse axis

*b* = Length of semi-conjugate axis.