Complex Numbers
Intersection of algebraic curves in complex form
MJAT_TS4_P1
Grade 12
Question:
Curves $A$, $B$, $C$, $D$ in the plane are defined as:
$A=\{(x,y): x^2-y^2;\; x^2y-y^3\}$, $B=\{(x,y): 2xy;\; x^3-3xy^2+3y=1\}$,
$C=\{(x,y): x^2-y^2+y=3x\}$, $D=\{(x,y): 3x^2y-y^3=0\}$.
Then:
A) $(C\cap A)=(C\cap B)$
B) $(C\cap A)\neq(C\cap B)$
C) $n(C\cap D)=2$
D) $n(C\cap D)=4$
Step-by-Step Solution
Key Concept: Recognize $A$, $B$, $C$, $D$ in terms of $z=x+iy$: $A$ is $\mathrm{Re}(z^2)$, $B$ is $\mathrm{Im}(z^3)=1$... $C$ is $\mathrm{Re}(z^2)+\ldots$, $D$ is $\mathrm{Im}(z^3)=0$. Use complex number identities to find intersections.
Answer: **A**.
Correct Answer: A