How many times do the graphs of x^2 + y^2 + 4x + 6y = 12 and 5y = 4x + 13 intersect?

Guest Jan 4, 2020

#1**+2 **

Hello!

First note that the first equation is clearly a circle. We can complete the square on the first equation to get (x+2)^2+(y+3)^2=25 This is a circle with a center (-2,-3) and radius 5. The second equation can be expressed as y=4x/5 +13/5. After doing a quick graph, we can clearly see that there is 2 intersections.

USAMO2019 Jan 4, 2020