Question

Find the centre, eccentricity, foci and directrices of the hyperbola x² - y2 + 4x=0.

Answer 1

Step-by-step explanation:

Given equation of hyperbola is x

2

−3y

2

+6x+6y+18=0

(x

2

+6x)−(3y

2

−6y)=−18

(x

2

+6x+9−9)−3(y

2

−2y+1−1)=−18

(x+3)

2

−9−3(y−1)

2

+3=−18

3(y−1)

2

−(x+3)

2

=12

(÷ by 12)

4

(y−1)

2

−

12

(x+3)

2

=1; Centre (−3,1)

Transverse axis is parallel to y-axis

a

2

=4,b

2

=12

e=

1+

a

2

b

2

=

1+

4

12

=

1+3

=2

a=2,e=2

ae=2×2=4;e=2

Centre=(−3,1); Foci=(0,±ae)=(0,±4)

(i.e.,) F

1

=(0,4)+(−3,1)=(−3,5)

F

2

=(0,−4)+(−3,1)=(−3,−5)

Vertices (0,±a)=(0,±2)

(i.e.,) V

1

=(0,2)+(−3,1)=(−3,3)

and V

2

=(0,−2)+(−3,1)=(−3,−1)