The point $(x,y)$ is given by $x=a$(1 + cosθ) and $y=a$·sinθ as θ varies. Which equation does it always satisfy?

Isolate the ratios first: $x-a=a$·cosθ and $y=a$·sinθ. Squaring and adding gives $a^2$(co$s^2$θ + si$n^2$θ) = $a^2$. The point therefore traces a circle of radius a centred at (a, 0), not at the origin — dropping the shift gives $x^2+y^2=a^2$, which fails at θ = 0° where the point is $(2a,0)$.