A grandfather and his granddaughter have their ages in the ratio 9 : 2. Sum of their ages is a prefect square. The difference in their ages is a multiple of 11. What are their ages?

Let say, grandfather and his granddaughter's ages are 9x and 2x.

So,

$$9x+2x=k^2.\ \left(k^2\ is\ a\ perfect\ square\ term\right)$$

or,$$11x=k^2.$$

And,

$$9x-2x=11m.\ \left(11m\ is\ a\ multiple\ of\ 11\right)$$

or,$$7x=11m.$$

or,$$x=\frac{11m}{7}.$$

So, $$\frac{11\times11m}{7}=k^2.$$

So, k would be a perfect square when m=7,28,63....

But,from the choice we can say that m=7,

then $$\frac{11\times11\times7}{7}=k^2.$$

or,$$k=11.$$

So,$$x=11.$$

So, their ages were 99 and 22.

A is correct choice.