A is a circle around the origin and a line through the origin, so they'll intersect in two places; we can skip the detailed calculation.
B is the intersection of a parabola and a line; they can intersect in zero, one or two places depending on the details. Let's check
[tex]y=x^2-7x+10=-6x+5[/tex]
[tex]x^2-x+5= 0[/tex]
Discriminant,
[tex]d = b^2-4ac=1-(4)(5)=-19[/tex]
Negative discriminant, zero real solutions for B.
System C is again a parabola and a line; we proceed similarly:
[tex]y = 8x+17 = -2x^2+9[/tex]
[tex]2x^2+8x+8=0[/tex]
[tex]x^2 + 4x + 4 = 0[/tex]
[tex]d=b^2-4ac=16 - 4(1)(4)=0[/tex]
Zero discriminant, exactly one real solution for C.
