How do you find the exact solutions to the system #3x^2-20y^2-12x+80y-96=0# and #3x^2+20y^2=80y+48#?

1 Answer
A. S. Adikesavan
Dec 24, 2016

#(x, y) = (-4, 0), (-4, 4), (6, 1) and (6, 3)#. A nice Socratic graph illustrates the crossings of the hyperbola and the ellipse represented by the given equations.

Explanation:

Add and subtract.

#x^2-2x-24=0#, giving #x -4 and 6#.

10y^2-40y+12+3x=0#, giving,

at #x = -4, y(y-4)=0 that gives common points (-4, 0) and (-4, 4) and ,

at #x = 6, y^2-4y+3= 0#, giving the common points (6, 1) and (6, 3).

The given equations represent a hyperbola and an ellipse,

respectively

graph{(3x^2-20y^2-12x+80y-96)(3x^2+20y^2-80y-48)=02 [-10, 10, -5, 5]}

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