The locus of the point of intersection of the straight lines tx-2y-3t=0 and x-2ty+3=0 (t R…

Question

The locus of the point of intersection of the straight lines $tx-2y-3t=0$ and $x-2ty+3=0$ ($t\in\mathbb R$), is:

Accepted Answer

From each equation: $t=\dfrac{2y}{x-3}$ and $t=\dfrac{x+3}{2y}$. Equating:
$$4y^2=(x-3)(x+3)=x^2-9\ \Longrightarrow\ \frac{x^2}9-\frac{y^2}{9/4}=1.$$

A hyperbola with $b^2=\dfrac94\Rightarrow$ conjugate axis $=2b=3$.

Solutions · 1