CONIC SECTIONS

CONIC SECTIONS

HYPERBOLA A hyperbola is the set of all points in a plane, the difference of whosedistances from two fixed points in the plane is a constant.

CONIC SECTIONS

HYPERBOLA The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola. The line through the foci is called the transverse axis and the line through the centre and perpendicular to the transverse axis is called the conjugate axis. The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola

The length of the transverse axis by 2a, the length of the conjugate axis by 2b and the distance between the foci by 2c. Thus, the length of the semi transverse axis is a and semi-cojugate axis is b CONIC SECTIONS

HYPERBOLA

From the figure b = √ c = a2 + b 2

√

c 2 − a2

CONIC SECTIONS

GENERAL EQUATION

CONIC SECTIONS

Eccentricity The eccentricity of hyperbola is the ratio of the distances from the centre of the hyperbola to one of the foci and to one of the vertices of the hyperbola (eccentricity is denoted by e) i.e., e = ca Latus rectum Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola. l=

2b 2 a

CONIC SECTIONS

Note if Equation of hyperbola is in the form

x2 a2

−

y2 b2

=1

y2 a2

−

x2 b2

=1

Focus is on the x-axis (±c, 0) Vertices=(±a, 0) Length of transverse axis=2a Length of cojugate axis=2b Length of latus rectum =

2b 2 a

if Equation of hyperbola is in the form Focus is on the y-axis (0, ±c) Vertices=(0, ±a) Length of transverse axis=2a Length of cojugate axis =2b Length of latus rectum =

2b 2 a

CONIC SECTIONS

QUESTION01 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus 2 x2 − y9 = 1 rectum of the hyperbola 25 ANSWER01 Comparing the given equation with

x2 a2

−

y2 b2

a=5,b=3 √ √ √ c = a2 + b 2 = 25 + 9 = 34 √ focii = (± 34, 0) Vertices = (±5, 0) Length of transverse axis=2a=10 Length of conjugate axis=2b=6 e=

c a

√

=

34 5

Length of latus rectum =

2b 2 a

=

2.9 5

=

18 5

CONIC SECTIONS

=1

QUESTION02 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus 2 y2 rectum of the hyperbola x9 − 16 =1 ANSWER02 Comparing the given equation with

x2 a2

−

y2 b2

a=3,b=4 √ √ √ c = a2 + b 2 = 9 + 16 = 25 = 5 focii = (±5, 0) Vertices = (±3, 0) Length of transverse axis=2.3=6 Length of conjugate axis=2b=8 e=

c a

=

5 3

Length of latus rectum =

2b 2 a

=

2.16 3

=

32 3

CONIC SECTIONS

=1

QUESTION03 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus 2 y2 =1 rectum of the hyperbola x4 − 25 ANSWER03 Comparing the given equation with

x2 a2

−

y2 b2

a=2,b=5 √ √ √ c = a2 + b 2 = 25 + 4 = 29 √ focii = (± 29, 0) Vertices = (±2, 0) Length of transverse axis=2a=4 Length of conjugate axis=2b=10 e=

c a

√

=

29 2

Length of latus rectum =

2b 2 a

=

2.25 2

= 25

CONIC SECTIONS

=1

QUESTION04 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus 2 x2 rectum of the hyperbola y9 − 27 =1 ANSWER04 2

Comparing the given equation with ya2 − √ a = 3, b = 3 3 √ √ √ c = a2 + b 2 = 9 + 27 = 36 = 6

x2 b2

focii = (0, ±6) Vertices = (0, ±3) Length of transverse axis=2.3=6 √ √ Length of conjugate axis=2.3 3 = 6 3 e=

c a

=

6 3

=2

Length of latus rectum =

2b 2 a

=

2.27 3

= 18

CONIC SECTIONS

=1

QUESTION05 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus 2 x2 rectum of the hyperbola 16 − y9 = 1 ANSWER05 Comparing the given equation with

x2 a2

−

y2 b2

a=4,b=3 √ √ √ c = a2 + b 2 = 16 + 9 = 25 = 5 focii = (±5, 0) Vertices = (±4, 0) Length of transverse axis=2a=8 Length of conjugate axis=2b=6 e=

c a

=

5 4

Length of latus rectum =

2b 2 a

=

2.9 4

=

9 2

CONIC SECTIONS

=1

QUESTION06 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus y2 x2 − 36 =1 rectum of the hyperbola 49 ANSWER06 Comparing the given equation with

x2 a2

−

y2 b2

a=7,b=6 √ √ √ c = a2 + b 2 = 49 + 36 = 85 √ focii = (± 85, 0) Vertices = (±7, 0) Length of transverse axis=2a=14 Length of conjugate axis=2b=12 e=

c a

√

=

85 7

Length of latus rectum =

2b 2 a

=

2.36 7

=

72 7

CONIC SECTIONS

=1

QUESTION07 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus y2 x2 − 100 =1 rectum of the hyperbola 25 ANSWER07 Comparing the given equation with

x2 a2

−

y2 b2

=1

a=5,b=10 √ √ √ √ c = a2 + b 2 = 25 + 100 = 125 = 5 5 √ focii = (±5 5, 0) Vertices = (±5, 0) Length of transverse axis=2a=10 Length of conjugate axis=2b=20 √ √ e = ca = 5 5 5 = 5 Length of latus rectum =

2b 2 a

=

2.100 10

= 20

CONIC SECTIONS

QUESTION08 Find the coordinates of the foci, the vertices, the length of transverse axis, the conjugate axis, the eccentricity and the latus rectum of the hyperbola 36x 2 − 4y 2 = 144 ANSWER08 36x 2 − 4y 2 = 144 divide through out by 144 x2 4

−

y2 36

=1

Comparing the given equation with

x2 a2

−

y2 b2

a=2,b=6 √ √ √ √ c = a2 + b 2 = 22 + 62 = 40 = 2 10 √ focii = (±2 10, 0) Vertices = (±2, 0) Length of transverse axis=2a=4 Length of conjugate axis=2b=12 CONIC SECTIONS

=1

ANSWER08..... e=

c a

=

√ 2 10 2

=

√

10

Length of latus rectum =

2b 2 a

=

2.36 6

= 12

CONIC SECTIONS