Question

Find the coordinates of the foci and the vertices, the eccentricity and the length of the latus rectum of the hyperbolas.`49 y^2-16 x^2=784`

Answer 1

Given equation is,
`49y^2-16x^2 = 784`
We will convert this equation into standard form.
`=>49/784y^2-16/784x^2 = 1`
`=> y^2/16-x^2/49 =1 `
So, this is our standard equation of hyperbola with,
`a = 4, b = 7`
`c = sqrt(a^2+b^2) = sqrt(16+49) = sqrt65`
Here, as `y^2` is positive, major-axis will be `Y-`axis.
Now, foci will be `(0,+-c) = (0,+-sqrt65).`
Vertices will be `(0,+-a) = (0,+-4).`
Eccentricity `= c/a = sqrt65/4.`
Length of latus rectum ` = 2b^2/a = 2**49/4 = 49/2.`