1. The graphs of these two equations intersect in how many points?
x^2 + y^2 =25
x^2- y^2 = 3
Add these equations and we get that
2x^2 = 28 divide through by 2
x^2 = 14 take both roots
x = sqrt (14) or x = -sqrt (14)
When x = sqrt (14) we have
(sqrt 14)^2 + y^2 = 25
14 + y^2 = 25
y^2 = 25 - 14
y^2 = 11
y = sqrt (11) or y = -sqrt (11)
So.......two points of intersection are (sqrt 14, sqrt 11) and ( sqrt 14, - sqrt 11)
Due to symmetry......the graphs will also intersect at (-sqrt 14, sqrt 11) and (-sqrt 14 , - sqrt 11)
So.....4 points of intersection
See the graph here : https://www.desmos.com/calculator/ie0t00e5nl