Question

How do you solve the system `x^2-y^2-8x+8y-24=0` and `x^2+y^2-8x-8y+24=0`?

Answer 1

The equations represent cencentric rectangular hyperbola of major axis `2sqrt 56` and a circle of lesser diameter `2sqrt8`, and so, there is no common point to give a solution.r

Explanation:

The first equation is

`(x-4)^2/56-(y-4)^2/56 = 1` representing a rectangular hyperbola

with center at (4, 4) and major axis `2sqrt 56`

The second has the form

(x-4)^2+(y-4)^2= 8# representing a circle with same center (4, 4) and

diameter `2sqrt 8 < 2sqrt 56`. As there is no intersection, there is

no solution..

When solved algebraically, we would be missing the graphical

aspect of the problem.

Answer 2

See below

Explanation:

Given

#{ (x^2 - y^2 - 8 x + 8 y - 24 = 0), (x^2 + y^2 - 8 x - 8 y + 24 = 0) :}#

Adding both sides we obtain

`2x^2-16x=0`. Solving for `x` we get at

`{x = 0, x = 8}`

Now substracting term to term the afore mentioned equations we obtain

`-2y^2+16y-48=0`

Solving for `y` we obtain

`y = 4 pm 2 i sqrt(2)` two complex conjugate roots. So the system dont have a real solution but has complex solutions

#( (x = 0, y = 4 - 2 i sqrt[2]), (x = 0, y = 4 + 2 i sqrt[2]), (x = 8, y = 4 - 2 i sqrt[2]), (x = 8, y = 4 + 2 i sqrt[2]))#