2022/024) Given $a_1^2+5a_2^2=10,\,a_2b_1-a_1b_2=5$ and $a_1b_1+5a_2b_2=\sqrt{105}$ for $a_1,\,a_2,\,b_1,\,b_2\in R$, evaluate $b_1^2+5b_2^2$.

we have

$(p^2+q^2)(l^2 + m^2) = (pl-qm)^2+ (pm + ql)^2$




putting $p= a_1$, $q = \sqrt{5} a_2$ ,$l= b_1$, $m = \sqrt{5} b_2$



we get $(a_1^2 + 5a_2^2)(b_1^2+5b_2)^2 = (a_1b_1 + 5a_2b_2)^2+5(a_1b_2 - a_2b_1)^2$



putting values from given conditions we get



$10(b_1^2+5b_2^2) = 105 + 5 * 5^2$

or $10(b_1^2 + 5b_2^2) = 230$

or $b_1^2 + 5b_2^2 = 23$