Which hyperbola has the smallest distance between the vertex and the focus belonging to the same branch

The hyperbola has the smalles distance between the vertex and the focus belonging to the same branch is 2.42:

Option B. (x-1)^2/12^2-(y-11)^2/8^2=1

Step-by-step explanation:

Form 1: (x-h)^2/a^2-(y-k)^2/b^2=1

Form 2: (y-k)^2/a^2-(x-h)^2/b^2=1

Distance between the vertex and the focus: c-a

c^2=a^2+b^2

A. (x+10)^2/11^2-(y-3)^2/10^2=1

a^2=11^2→a=11

b^2=10^2

c^2=a^2+b^2→c^2=11^2+10^2→c^2=121+100→c^2=221→sqrt(c^2)=sqrt(221)→

c=14.87

c-a=14.87-11→c-a=3.87

B. (x-1)^2/12^2-(y-11)^2/8^2=1

a^2=12^2→a=12

b^2=8^2

c^2=a^2+b^2→c^2=12^2+8^2→c^2=144+64→c^2=208→sqrt(c^2)=sqrt(208)→

c=14.42

c-a=14.42-12→c-a=2.42

C. (y-8)^2/9^2-(x+6)^2/12^2=1

a^2=9^2→a=9

b^2=12^2

c^2=a^2+b^2→c^2=9^2+12^2→c^2=81+144→c^2=225→sqrt(c^2)=sqrt(225)→

c=15

c-a=15-9→c-a=6

D. (y-14)^2/9^2-(x+28)^2/11^2=1

a^2=9^2→a=9

b^2=11^2

c^2=a^2+b^2→c^2=9^2+11^2→c^2=81+121→c^2=202→sqrt(c^2)=sqrt(202)→

c=14.21

c-a=14.21-9→c-a=5.21

The smallest distance between the vertex and the focus belonging to the same branch (c-a) is 2.42 (Option B.)

true

step-by-step explanation:

one set of data has more variation than the other. hope this !