Đáp án:`S=12`

Gọi toạ độ `M(x_1;y_1),N(x_2;y_2)`

`=>x_1=HM;x_2=HN`

mà `HM=2MN =>HN=3MN`

`=>x_1/x_2=(HM)/(HN)=2/3`

`=>x_1=2/3 x_2`

Ta có `y_1=y_2`

hay `a^x_1 =b^x_2`

mà `a=bsqrt2; x_1=2/3x_2`

`=>(bsqrt2)^(2/3x_2)=b^x_2`

`<=>(bsqrt2)^(2/3)=b`

`<=>2/3=log_(bsqrt2)b`

`<=>log_b(bsqrt2)=3/2`

`<=>log_bsqrt2+1=3/2`

`<=>log_bsqrt2=1/2`

`=>b=2`

`=>a=sqrt2 b=2sqrt2`

`=>S=a^2+b^2=2^2+(2sqrt2)^2=12`