Equation of a conic section and describe it?

We have

4x² - y² - 8x - 2y - 1 = 0

( 4x² - 8x ) - ( y² + 2y + 1 ) = 0

4( x² - 2x ) - ( y + 1 )² = 0

4( x² - 2x + 1 ) - 4 - ( y + 1 )² = 0

4( x - 1 )² - ( y + 1 )² = 4

Dividing both sides by 4,

[ ( x - 1 )² ] - [ ( y + 1 )² / 4 ] = 1

( X² / 1² ) - ( Y² / 2 ² ) = 1, ............ where ... X = x - 1, ... Y = y + 1

................................................................................................................................................

This equation represents a Hyperbola.

Its center is ≡ ( X=0, Y=0 ) ≡ ( x-1=0, y+1=0 ) ≡ ( 1, -1 ),

semi-transverse axis = a = 1

semi-conjugate axis = b = 2.

.................................................................................................................................................

Happy To Help !

..................................................................................................................................................

The center is x=1 y=-1

Making the translation to this point

X=x-1

Y=y+1

4(X+1)^2-((Y-1)^2-8X+8-2Y+2-1=0

4X^2+8X+4 -Y^2+2Y-1-8X+8-2Y+2-1=0

so 4X^2-Y^2 =-12 so Y^2/12-X^2/3 =1 It´s a Hyperbole With it´s foca axis parallel to Oy

a^2=12 and b^2 =3

Check numbers