Q.A variable straight line passes through the point of intersection of straight lines x/a+y/b=1 and x/b+y/a=1 and intersects the axes at P and Q. Find the locus of mid-point of PQ.

Find the fixed point of intersection, write the variable line in intercept form through it, then eliminate the intercepts using the midpoint relation.

Fixed point of intersection of $\dfrac xa+\dfrac yb=1$ and $\dfrac xb+\dfrac ya=1$: subtracting, $(x-y)\left(\dfrac1a-\dfrac1b\right)=0$, and since $a\ne b$, $x=y$. Substituting into the first equation: $x\left(\dfrac1a+\dfrac1b\right)=1\implies x=\dfrac{ab}{a+b}$. So the fixed point is $I=\left(\dfrac{ab}{a+b},\dfrac{ab}{a+b}\right)$.

Variable line through $I$, with intercepts $P=(p,0),Q=(0,q)$: $\dfrac Xp+\dfrac Yq=1$. Since it passes through $I$:

$\dfrac{ab/(a+b)}{p}+\dfrac{ab/(a+b)}{q}=1\implies\dfrac1p+\dfrac1q=\dfrac{a+b}{ab}.$

…