Form a differential equation of the family of all hyperbola whose center at (2,1).

Question

Please I need your help ASAP. Thank you! 

Answer 1

The standard equation of a  vertical hyperbola whose center is at (2,1) is

 

(y-1)^2/a^2 - (x-2)^2/b^2 = 1

 

multiplying each side by a^2*b^2  gives

 

b^2(y-1)^2 - a^2(x-2)^2 = a^2*b^2

 

We also have a horizontal hyperbola whose form would be

 

a^2(x-2)^2 - b^2(y-1)^2 = a^2*b^2

 

We can generalize these equations by

 

c1*(y-1)^2 - c2*(x-1)^2 = c3,  where c1, c2 and c3 are constants not equal to zero

 

Taking two derivatives implicitly we obtain

 

eq1) 2c1*(y-1)y' - 2c2*(x-2) = 0

eq2) 2c1*(y-1)y'' - 21c1*y' - 2c2 = 0

 

Solve eq1 for c2 and substitute into that result for c2 into eq2.  Now divide through by c1 and simplify.  I will leave the rest for you to finish.  Any questions just ask.