The point of intersection of the lines (a^3+3)x+ay+a-3=0 and (a^5+2)x+(a+2)y+2a+3=0 (a rea…

Question

The point of intersection of the lines $(a^3+3)x+ay+a-3=0$ and $(a^5+2)x+(a+2)y+2a+3=0$ (a real) lies on the $y$-axis for:

Accepted Answer

Setting $x=0$ in both: $ay+a-3=0$ and $(a+2)y+2a+3=0$.

For a common solution: $(a+2)(a-3) - a(2a+3) = 0$
$$a^2-a-6-2a^2-3a=0 \implies -a^2-4a-6=0 \implies a^2+4a+6=0$$

Discriminant $=16-24=-8<0$: no real solutions.

Solutions · 1