Question 32650
ASYMPTOTES ARE
Y=2X…OR….Y-2X=0……I………………..AND
Y=-2X…OR….Y+2X=0……II…...SO THEIR COMBINED EQN.IS GIVEN BY EQN.I*EQN.II=0
(Y-2X)(Y+2X)=0=Y^2-4X^2=0………EQN.3.
IT IS KNOWN THAT THE EQN.OF HYPERBOLA DIFFERS FROM THE COMBINED EQN.OF ASYMPTOTES ONLY BY A CONSTANT.
SO…EQN.OF HYPERBOLA IS ….
Y^2-4X^2=C……..III…………...WHERE C IS A ONSTANT TO BE FOUND…PUTTING THIS EQN.IN STD.FORM WE GET
Y^2/C -4X^2/C =1….OR…..
(Y-0)^2/C - (x-0)^2/(C/4) =1…………………………………………IV
COMPARING WITH STANDARD EQN.OF HYPERBOLA…						
(Y-K)^2/B^2-(X-H)^2/A^2=1.………………………………………V						
WHERE VERTICES ARE...(H,(K-B)) AND (H,(K+B)) …GIVEN HERE AS (0,-5) AND (0,5)						
WE GET H=0						
K+B=5						
K-B=-5						
ADDING THE ABOVE 2 EQNS…..2K=0…OR…K=0						
SO…B=5						
COMPARING  EQN.IV,WITH STD.EQN.V,WE GET….						
H=0…..K=0…….C=B^2……..C/4=A^2…….BUT B=5 WE FOUND .SO						
C=5^2=25…….A^2=25/4						
HENCE EQN.OF HYPERBOLA IS AS PER EQN.III IS 						
Y^2-4X^2=25