\(\Rightarrow \text { In 't' time , } \Delta Q =\left(\frac{ V ^{2}}{ R }\right) t\)

Given that, (for same source, \(v=\) same)

\(Q _{0}=\frac{ v ^{2}}{ R _{1}} \times 20=\frac{ V ^{2}}{ R _{2}} \times 60 \ldots\) \((i)\)

\(\Rightarrow R _{2}=3 R _{1} \ldots.\) \((ii)\)

If they are connected in parallel then

\(\operatorname{Req}=\frac{ R _{2} R _{1}}{ R _{1}+ R _{2}}=\frac{3 R _{1} \cdot R _{1}}{3 R _{1}+ R _{1}}=\left(\frac{3 R _{1}}{4}\right)\)

To produce same heat, using equation ...\((i)\)

\(Q _{0}=\frac{ V ^{2}}{ R _{1}} \times 20=\frac{ v ^{2}}{\left(\frac{3 R _{1}}{4}\right)} \times t\)

\(t =\frac{3 \times 20}{4}=15 \,min\)