Question

3x^2 + y^2 + 2z^2 + 3x + 4y + 4z= 0 represent ellipsoid. find its center and length of semi-axis

Answer 1

Step-by-step explanation:

Equation of an ellipse is 3x

2

+4y

2

−6x+8y−5=0

∴3x

2

−6x+4y

2

+8y−5=0

⇒3x

2

−6x+3−3+4y

2

+8y+4−4−5=0

⇒3x

2

−6x+3+4y

2

+8y+4−12=0

⇒3(x

2

−2x+1)+4(y

2

+2y+1)−12=0

⇒3(x

2

−2x+1)+4(y

2

+2y+1)=12

⇒

4

(x

2

−2x+1)

+

3

(y

2

+2y+1)

=1

⇒

4

(x−1)

2

+

3

(y+1)

2

=1

Comparing it with ⇒

a

2

(x−h)

2

+

b

2

(y−k)

2

=1

We get,

h=1,y=−1,a=2,b=

3

So center of ellipse is (1,−1),

For eccentricity e,

e=

1−

a

2

b

2

e=

1−

4

3

∴e=

2

1