 Hello and welcome to the session. In this session we discussed the following question which says if q is the mean proportional between p and r, prove that p square minus q square plus r square is equal to q to the power 4 into 1 upon p square minus 1 upon q square plus 1 upon r square the whole. We know that if three quantities a, b and c are in continued proportion that is a is to b is equal to b is to c then this b is called the mean proportional between a and c. This is the key idea that we use for this question. Let's proceed with the solution now. Now that we have that q is the mean proportional between p and r therefore p is to q is equal to q is to r or you can say p upon q is equal to q upon r. We take each of these ratios equal to k so when p upon q is equal to k this means that p is equal to qk and when q upon r is equal to k this means q is equal to rk. Now that q is equal to rk so we put q as rk in this so we have p is equal to rk into k and this is equal to rk square that is we have p equal to rk square and q equal to rk. Now we are supposed to prove that p square minus q square plus r square is equal to q to the power 4 into 1 upon p square minus 1 upon q square plus 1 upon r square the whole. Now consider the rhs. Rhs is q to the power 4 into 1 upon p square minus 1 upon q square plus 1 upon r square the whole. Now substituting p equal to rk square and q equal to rk we get rk this whole to the power 4 into 1 upon p square that is rk square the whole square minus 1 upon q square that is 1 upon rk whole square plus 1 upon r square the whole. So this is further equal to r to the power 4 into k to the power 4 into 1 upon r square k to the power 4 minus 1 upon r square k square plus 1 upon r square the whole. Now here we have r to the power 4 into k to the power 4 upon r square we have taken this r square common from these denominators and this into 1 upon k to the power 4 minus 1 upon k square plus 1. Now here this r square cancels with r square so here we are left with r square into k to the power 4. Now taking LCM of the denominators inside the brackets we have k to the power 4 in the denominator and in the numerator we have 1 minus k square plus k to the power 4. Then this k to the power 4 and k to the power 4 cancels so we are left with r square into k to the power 4 minus k square plus 1 the whole. Or we can rewrite this as r square into k to the power 4 minus r square k square plus r square that is we can say r k square the whole square minus r k the whole square plus r square. Now we know that q is equal to rk and p is equal to rk square so in place of rk we put q and in place of rk square we put p so this would be equal to p square minus q square plus r square since we know p is equal to rk square q is equal to rk and this is equal to the LHS that is this that we get the RHS is equal to the LHS hence we can say that p square minus q square plus r square is equal to q to the power 4 into 1 upon p square minus 1 upon q square plus 1 upon r square the whole. So this is what we were supposed to prove this completes the session hope you have understood the solution of this question.