Question 1180216
Which hyperbolas have one vertex in common with the hyperbola

{{{(y-4)^2/7^2-(x+6)^2/8^2=1}}} 

Up-down hyperbola with  center ({{{h}}}, {{{k}}} )= ({{{-6}}}, {{{4}}}), {{{a=7}}}, {{{b=8}}}

The vertices: 

({{{h}}}, {{{k+a}}} ), ({{{h}}}, {{{k-a}}} )
=>({{{-6}}}, {{{4+7}}} ), ({{{-6}}}, {{{4-7}}} )
 =>({{{-6}}},{{{ 11}}} ), ({{{-6}}}, {{{-3}}} )



a.

{{{ (y-8)^2/5^2-(x+6)^2/6^2=1}}} 

 center ({{{h}}}, {{{k}}} )= ({{{-6}}}, {{{8}}}), {{{a=5}}}, {{{b=5}}}
The vertices: 
({{{h}}}, {{{k+a}}} ), ({{{h}}}, {{{k-a}}} )
=>({{{-6}}}, {{{8+5}}} ), ({{{-6}}}, {{{8-5}}} )
 =>({{{-6}}},{{{ 13}}} ), ({{{-6}}}, {{{3}}} )


b. 


{{{(x-1)^2/8^2-(y-11)^2/7^2=1}}} 

Right-left hyperbola with  ({{{h}}},{{{ k}}} )= ({{{1}}}, {{{11 }}}), {{{a=8}}}, {{{b=7}}}

The vertices 
({{{h+a}}}, {{{k }}}), ({{{h-a}}},{{{ k}}} )
=>({{{1+8}}}, {{{11 }}}), ({{{1-8}}},{{{ 11}}} )
=>({{{9}}}, {{{11 }}}), ({{{-7}}},{{{ 11}}} )


c. 

{{{(x+15)^2/9^2-(y+3)^2/5^2=1}}}

Right-left hyperbola with  ({{{h}}},{{{ k}}} )=  ({{{-15}}}, {{{-3 }}}), {{{a=9}}}, {{{b=5}}}
The vertices 
({{{h+a}}}, {{{k }}}), ({{{h-a}}},{{{ k}}} )
=>({{{-15+9}}}, {{{-3 }}}), ({{{-15-9}}},{{{ -3}}} )
=>({{{-6}}}, {{{-3 }}}), ({{{-24}}},{{{ -3}}} )


d. 


{{{(y-6)^2/6^2-(x+6)^2/4^2=1}}}

Up-down hyperbola with  center ({{{h}}}, {{{k}}} )= ({{{-6}}}, {{{6}}}), {{{a=6}}}, {{{b=4}}}

The vertices: 
({{{h}}}, {{{k+a}}} ), ({{{h}}}, {{{k-a}}} )
=>({{{-6}}}, {{{6+6}}} ), ({{{-6}}}, {{{6-6}}} ) 
=>({{{-6}}},{{{ 12}}} ), ({{{-6}}}, {{{0}}} )


e. 


{{{(y-23)^2/12^2-(x+6)^2/8^2=1}}} 

Up-down hyperbola with  center ({{{h}}}, {{{k}}} )= ({{{-6}}}, {{{23}}}), {{{a=12}}}, {{{b=8}}}

The vertices: 
({{{h}}}, {{{k+a}}} ), ({{{h}}}, {{{k-a}}} )
=>({{{-6}}}, {{{23+12}}} ), ({{{-6}}}, {{{23-12}}} )
 =>({{{-6}}},{{{ 35}}} ), ({{{-6}}}, {{{11}}} )


hyperbolas which have one vertex in common with the given hyperbola:

c has =>({{{-6}}}, {{{-3 }}})
e has =>({{{-6}}}, {{{11}}} )