∵ The focus is on the X axis, ∴ Let the hyperbolic equation be: X 2/A 2-Y 2/B 2 = 1.
∫ real major axis =8, ∴2a=8,a=4.
∫ imaginary axis length =2, ∴2b=2,b= 1.
The hyperbolic equation is: x 2/ 16-y 2 = 1.
(2) The focal length is 26 and passes through point M (0, 12)
When the focus is on the x axis, let the hyperbolic equation be: X 2/A 2-Y 2/B 2 = 1.
Focal length =26, ∴2C=26,c= 13.
∴a^2+b^2=c^2= 169
If m (0, 12) is brought into the hyperbolic equation, it is-12 2/b 2 = 1, ∫B2 > 0, which is not valid.
∴, the focus is not on the x axis,
When the focus is on the y axis, let the hyperbolic equation be: Y 2/A 2-X 2/B 2 = 0.
Steps are the same as above. After m is brought in, it is: 12 2/A 2 = 1, ∴ A = 12.
∵a^2+b^2=c^2= 169,∴b=5
Hyperbolic equation: y 2/144-x 2/25 =1.
(3) The focus is on the X axis, and the focal length is 10. The absolute value of the difference between the point m on the hyperbola and the distance between the two focal points is equal to 6.
∵ The focus is on the X axis, ∴ Let the hyperbolic equation be: X 2/A 2-Y 2/B 2 = 1.
Focal length = 10,∴2c= 10,c=5,
The absolute value of the difference between the point m on the hyperbola and the distance between the two focal points is equal to 6.
∴2a=6,a=3
∵a^2+b^2=c^2,∴b=4
The hyperbolic equation is: x 2/9-y 2/ 16 = 1.