Question

Find the equations of the hyperbolas that intersect `3x^2-4y^2=5xy` and `3y^2-4x^2=2x+5`?

Find the equations of the hyperbolas that intersect
`3x^2-4y^2=5xy` and `3y^2-4x^2=2x+5`

Answer 1

Given:

`{ (3x^2-4y^2=5xy), (3y^2-4x^2=2x+5) :}`

Note that the first of these "hyperbolas" is degenerate, being the union of two straight lines.

graph{(3y^2-4x^2-2x-5)(3x^2-4y^2-5xy) = 0 [-6, 6, -3, 3]}

Subtracting `5xy` from both sides of the first equation, we get:

`3x^2-5xy-4y^2 = 0`

Multiply through by `12`, then complete the square as follows:

`0 = 12(3x^2-5xy-4y^2)`

`color(white)(0) = 36x^2-60xy-48y^2`

`color(white)(0) = (6x)^2-2(6x)(5y)+(5y)^2-73y^2`

`color(white)(0) = (6x-5y)^2-(sqrt(73)y)^2`

`color(white)(0) = ((6x-5y)-sqrt(73)y)((6x-5y)+sqrt(73)y)`

`color(white)(0) = (6x-(5+sqrt(73))y)(6x-(5-sqrt(73))y)`

The other (proper) hyperbola intersects the line:

`y = 6/(5-sqrt(73))x`

Substitute this expression for `y` into the second equation to get:

`3(6/(5-sqrt(73))x)^2-4x^2 = 2x+5`

That is:

`(4-3(6/(5-sqrt(73)))^2)x^2+2x+5 = 0`

Hence:

`x = -(4 (sqrt(254 + 150 sqrt(73)) +- 8))/(19 + 15 sqrt(73))`

Then:

`y = 6/(5-sqrt(73))x`