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

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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

##### 1 Answer

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

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

Multiply through by

#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

#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#