The hyperbola equation is . \ \

The standard form of the equation of a hyperbola with center (h, k) (where a and b are not equals to 0) is (Transverse axis is horizontal) or (Transverse axis is vertical).

Write the hyperbola in the standard form.

\ \

The vertices and foci are, respectively a and c units from the center (h, k)  and the relation between a, b and c is b² = c² - a².

Compare the equation with .

a² = 4, b² = 2, k = - 3 and h = 1.

a = ± 2 and b = ± √2.

To find the value of c, substitute the value of a² = 4 and b² = 2 in b² = c² - a².

2 = c² - 4

2 + 4 = c²

c = ± √6.

1).

Here the transverse axis is vertical, the asymptotes , with center (h, k) are of the forms y - k = (a/b)(x - h) and y - k = - (a/b) (x - h).

Substitute the values of (h, k) = (1, - 3), a = ± 2 and b = ± √2  in y - k = ±(a/b)(x - h)

y - (- 3)= ± 2/√2 (x - 1) \ \

The asymptote equations are y + 3= ± √2 (x - 1).

The right choice is option c. \ \

2).

Foci = (h, k ± c).

Substitute the values (h, k) = (1, - 3) and c = ± √6.

Foci = (h, k ± c) = (1, - 3 ± √6,).

The right choice is option d.