Question

Rewrite the equation y^2 - 2x + 2y-5=0 in standard form. Determine the focus and directrix

Answer 1

Answer : option D

$y^2 - 2x + 2y-5=0$

We apply completing the square method

Move all the y terms on one side

y^2 +2y = 2x + 5

In completing the square method , we take coefficient of y then divide it by 2  and square it

2/2 =1  then 1^2 = 1

Add it on both sides

$y^2 + 2y +1 = 2x+5+1$

$(y+1)^2= 2x+6$

$(y+1)^2= 2(x+3)$

Solve for x

$(y+1)^2= 2x+6$

$2x=(y+1)^2-6$ (divide by 2 on both sides)

$x=1/2(y+1)^2-3$

Vertex is (h,k) that is (-3,-1). h= -3  and k = -1

The value of a= 1/2

$p = \frac{1}{4a}$

Plug in 1/2 for 'a'

so P = 1/2

Focus = (h+p, k)

h= -3  and k = -1, p = 1/2

$(-3+\frac{1}{2} , -1) = (\frac{-5}{2} , -1)$

Directrix x=(h - P)

$x = -3 - \frac{1}{2} = \frac{-7}{2}$

Option D is correct