Equations of Lines — Question 8

Question 8 – Solution

Strategy Distinct axis intercepts determine a line in the $xy$-plane, but both intercepts may be the origin.

Distinct intercepts A line through distinct points $X=(a,0,0)$ and $Y=(0,b,0)$ lies in $z=0$, so it cannot contain $P$.

Coincident intercepts The axes share the origin. The line through the origin and $P$ is $$\boxed{\vec r(t)=t\left\langle 1,2,3\right\rangle,\qquad t\in\mathbb R}.$$ It passes through $P$ at $t=1$ and meets each axis at $t=0$. Thus both intercepts are $\boxed{(0,0,0)}$.

Key idea The distinct-point argument does not rule out coincident intercepts. This is the unique line satisfying the question.