Two sides of an isosceles triangle are given by the equations $7x- y+3=0 and x+ y-3=0.$ If its third side passes through the point $(1, - 10),$ then its equations are

Any line through $(1, -10) is (y + 10) = m (x -1)$

Since it makes equal angles, say $\theta ,$with the given lines, we have 

$\tan \theta =\frac{m-7}{1+7m}=-\frac{m-(-1)}{1+m(-1)}$

This gives $m=1/3\text{ or }-3.$

Hence the equation of third side is 

$$\begin{gathered} y+10=\frac{1}{3}(x-1) \\ y+10=-3(x-1) i.e, x-3y-31=0 \\ \text{ or }3x+y+7=0 \end{gathered}$$