Show that the equation 9`x^2-16 y^2-18 x+32 y-151=0` represents a hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latus-rectum, coordinates of foci and vertices, equations of the directrices of the hyperbola.

Options

A. one of the directrix is `x=21//5`
B. the length of latus rectum is `9//2`
C. foci are (6, 1) and `(-4, 1)`
D. the eccentricity is 5/4

Answer

A, B, C, D

Explanation

The given hyperbola can be written as
`((x-1)^(2))/(16)-((y-1)^(2))/(9)=1`
`"or "(x^(2))/(16)-(y^(2))/(9)=1" (where X= x - 1, Y = y - 1)"`
`e=sqrt(1+(b^(2))/(a^(2)))=sqrt(1+(9)/(16))=(5)/(4)`
Directrices are `X= pma//e`. Therefore,
`x-1= pm (16)/(5)`
`"or "x=(21)/(5) and x=-(11)/(5)`
Length of latus rectum `=(2b^(2))/(a)=(9)/(2)`
The foci are given by
`X= pmae, Y = 0`
`"or "(6,1),(-4,1)`