Question

Ally is analyzing a circle, y2 + x2 = 4, and a linear function g(x). Will they intersect?

y2 + x2 = 4 g(x)
graph of the function y squared plus x squared equals 4
x g(x)
0 3
4 0
8 −3

Answer 1

the linear function gives us the points
(0,3)
(4,0)
(8,-3)

find the equation
hm
find slope
(y2-y1)/(x2-x1)
for (0,3) and (4,0)
slope=(0-3)/(4-0)=-3/4

y=-3/4x+b
given
(0,3)
y=-3/4x+3


so sub -3/4x+3 for y in other equation
(-3/4x+3)^2+x^2=4
expand
9/2x^2-9/2x+9+x^2=4
11/2x^2-9/2x+9=4
times 2 both sides
11x^2-9x+18=8
minus 8 both sides
11x^2-9x+10=0
factor
or use quadratic formula
for
ax^2+bx+c=0
x=
`(-b+/- √(b^2-4ac) )/(2a)`

11x^2-9x+10=0
x=
`(-(-9)+/- √((-9)^2-4(11)(10)) )/(2(11))`
x=
`(9+/- √(81-440) )/(22)`
x=
`(9+/- √(-359) )/(22)`
we see we have no real roots
there are no intersection points