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.)

the sum of all the angles in a triangle are equal to 180

so you can add   the 3 given angles together to equal 180:

5y + 2y + 2y = 180

or since all three angles have the same letter, combine the 3 angles to equal 180:

9y = 180

with btw u said 100pts but it clearly says

i answerd and just got 10pts

no hate