 Hello and welcome to the session. In this session we discussed the following question which says, if 4A squared plus 7B squared upon 4A squared minus 7B squared is equal to 4C squared plus 7D squared upon 4C squared minus 7D squared show that A upon B is equal to C upon D. Before moving on to the solution, first of all we will discuss a result on proportions which is componento dividendo. According to this we have, if 4 quantities AB and CR in proportion that is A is to B is equal to C is to D or you can say A upon B is equal to C upon D then on applying componento dividendo we have A plus B upon A minus B is equal to C plus D upon C minus D. This is the key idea that we use for this question. Let's move to the solution now. We are given that 4A squared plus 7B squared upon 4A squared minus 7B squared is equal to 4C squared plus 7D squared upon 4C squared minus 7D squared. Let this be result one and we are supposed to show that A upon B is equal to C upon D. Let us now apply componento and dividendo to result one so applying componento and dividendo to result one we get 4A squared plus 7B squared plus 4A squared minus 7B squared this whole upon 4A squared that is the numerator plus 7B squared minus the denominator so we get minus 4A squared plus 7B squared is equal to the numerator of the right hand side which is 4C squared plus 7D squared plus the denominator which is 4C squared minus 7D squared and this whole upon numerator 4C squared plus 7D squared minus the denominator so we get minus 4C squared plus 7D squared. Now in the numerator of the left hand side we have 7D squared minus 7D squared cancels then in the denominator 4A squared and minus 4A squared cancels then on the right hand side in the numerator 7D squared and minus 7D squared cancels then 4C squared minus 4C squared gets cancelled so we have 4A squared plus 4A squared it becomes 8A squared upon 7B squared plus 7B squared becomes 14B squared equal to 4C squared plus 4C squared becomes 8C squared upon 7D squared plus 7D squared becomes 14B squared. Now here this 8 cancels with this 8, 14 cancels with this 14 and we are left with A squared upon B squared is equal to C squared upon D squared. This means we have A upon B is equal to C upon D and this is what we were supposed to prove so hence proved that A upon B is equal to C upon D. This completes the session hope you have understood the solution of this question.