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.)

hello :

by applying the distributive property :

6 × 4.3 = 6 × ( 4 + 0.3 )

            =   (6 ×   4) + (6 ×   0.3)

            = 24 + 1.8

6 × 4.3 = 25.8