= Solution
In an affine chart, the <Zariski tangent space> at $P$ consists of vectors $v$ with $df_P(v)=0$ for every <polynomial> vanishing on the variety. Intrinsically it is the dual of $\mathfrak m_P/\mathfrak m_P^2$. For a finite set of generators in an ambient $N$-space, its dimension is $N-\operatorname{rank}J(P)$. The condition that this be at least $r$ is the vanishing of all $(N-r+1)$-minors, hence is closed. The affine-chart descriptions agree intrinsically; being closed is local on the variety, so the same conclusion holds projectively.
For the two given quadrics the gradient rows are
$$
J=\begin{pmatrix}0&X_2&X_1&X_4&X_3\\X_1&X_0&0&2X_3&2X_4\end{pmatrix}.
$$
Their intersection has dimension two: the quadrics have no common factor, so form a codimension-two complete intersection. Its singular points are where the two rows are dependent. If the first row vanishes, one obtains $[1:0:0:0:0]$. Otherwise write a dependence with nonzero coefficient of the second row. Its first coordinate forces $X_1=0$. The defining equations then imply $X_3X_4=0$ and $X_3^2+X_4^2=0$, hence $X_3=X_4=0$. Conversely every point of the resulting line lies on the variety and has dependent rows. Therefore
$$
\boxed{\operatorname{Sing}V=\{[X_0:0:X_2:0:0]\}\cong\mathbb P^1.}
$$
On this line at least one of $X_0,X_2$ is nonzero, so the Jacobian <rank> is exactly one. The projective ambient tangent dimension is four, giving \b[$\dim T_{V,P}=3$ at every singular point]. Away from the line the <rank> is two and the tangent dimension is two.
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