Identify correct statement(s) about conic `sqrt((x-5)^(2)+(y-7)^(2)) +sqrt((x+1)^(2)+(y+1)^(2)) =12`

Options

A. centre of conic is (2,3)
B. conic is hyperbola with foci (5,7) and `(-1,-1)`
C. conic is ellipse with major axis `4x -3y +1 =0`
D. eccentricity of conic is `(5)/(7)`

Answer

A, C

Explanation

`sqrt((x-5)^(2)+(y-7)^(2)) +sqrt((x+1)^(2)+(y+1)^(2))=12`
where `P(x,y), S(5,7)` and `S' (-1,-1)`.
`SP + S'P = 2a rArr` conic is ellipse
`2ae = sqrt(36+64)`
`2a = 12 rArr e = (5)/(6)`
Major axis `y - 7 = (7+1)/(5+1) (x-5)`
`rArr 4x -3y +1 =0`