1.29. Find the geometric locus of points $X$ such that $p A X^{2}+q B X^{2}+r C X^{2}=d$, where $A, B$ and $C$ are given points, $p, q, r$ and $d$ are given numbers, and $p+q+$ $+r=0$.
1.29. Let's introduce a coordinate system, directing the $C z$ axis perpendicular to the plane $A B C$. Suppose point $X$ has coordinates ($x$, $y$, $z$). Then, for example, $A X^{2}=\left(x-a_{1}\right)^{2}+\left(y-a_{2}\right)^{2}+z^{2}$. Therefore, for the coordinates of point $X$, we obtain an equation of the form ( $p+$ $+q+r)\left(x^{2}+y^{2}+z^{2}\right)+\alpha x+\beta y+\delta=0$, i.e., $\alpha x+\beta y+$ $+\delta=0$. This equation defines a plane perpendicular to the plane $A B C$ (in degenerate cases, it defines an empty set or the entire space).