be an element of the solution set of the system above. Find the equation of the line containing 
and 
for all such 
.
Benjamin H. answered • 11/17/21
Harvard Grad/Experienced Tutor in STEM, English, and Writing
So despite the conics involved, this actually boils down to algebra 2!
Since you have a system of equations, you can try and solve for each variable.
I chose to subtract the second equation from the first:
(y+2)^2 / 9 + (x-4)^2 / 25 = 1
-(y+2)^2 / 16 + (x-4)^2 / 25 = 1
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This left me with: (y+2)^2 / 9 + (y+2)^2 / 16 = 0.
By simplifying, you get that (y+2)^2 = 0
Thus, y = -2
From there, you can just substitute y=-2 back into your system of equations.
Doing so leaves you with:
(x-4)^2 / 25 = 1
For both equations. Thus, you know (x-4)^2 = 25, and that x-4 = +/- 5
Thus, x = 9 or -1
Now you have your solutions (a,b). (a,b) can be (9,-2) or (-1, -2).
From there you just take each of these points and find the equation of the line containing it and (-5, 4)
