= Solution
For a point $p\in\mathbb A_k^1$, let $a$ be the image of $t$ in its <residue field> $\kappa(p)$. The <scheme-theoretic fiber> is
$$
Y_p=\operatorname{Proj}\frac{\kappa(p)[x,y,z]}{(xy-az^2)}.
$$
If $p\ne(t)$, then $a\ne0$. The quadratic polynomial $xy-az^2$ is irreducible. Indeed, after setting $z=0$, any hypothetical linear factors must restrict, up to nonzero scalars, to $x$ and $y$; comparing the $xz$ and $yz$ coefficients then forces both $z$ coefficients to vanish, contradicting the nonzero $z^2$ coefficient. Hence its homogeneous coordinate ring is an <integral domain>, so $Y_p$ is an <integral scheme>.
At the origin $p=(t)$, the fiber is $V_+(xy)$, the union of the two distinct projective lines $x=0$ and $y=0$, and is therefore not irreducible. It is nevertheless a <reduced scheme> because the ideal $(xy)=(x)\cap(y)$ equals its <radical of an ideal>[radical]. Every other fiber is integral and hence reduced. Thus the fiber is integral exactly away from the origin, and it is reduced at every point of $\mathbb A_k^1$.
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