How to find smallest enclosing ellipse when multiple lines are given ? This ellipse needs to intersect all those lines

Question

I'm trying to find the smallest ellipse in terms of circumference. I suspect the smallest enclosing ellipse will intersect some lines in one single point. Given a line l: px + qy + r = 0 , L : {p q r}

I can try to find this ellipse by using the following equation: ap²_n + bp_nq_n + cq²_n + dp_nr_n + eq_nr_n + fr²_n = 0. Whereby n represents the number of lines. Using multiple lines I can find the values of a ... f.

Later I can use the following equation K: LBL^T= 0 , whereby B contains the parameters a ... f. I was actually wondering if there was a simpler or more elegant way to solve this problem.