13 If b is the meanproportional between a and c, prove that (ab + be)

1.	13 If b is the meanproportional between a and c, prove that (ab + be) is the meanproportional between (a2 + b2) and (b2 + c2).
Answer» here it is clearly given that b is the mean proportional between a and c . therefore, b^2 = ac Now (a ^2+b^ 2)(b ^2+c ^2) = (a ^2+ ac )( ac +c^ 2) , (a ^2+b^ 2)(b ^2+c ^2) = a(a + c ) c( a + c ) , (a ^2+b^ 2)(b ^2+c ^2) = a c (a + c )^2 , (a ^2+b ^2)(b^ 2+c ^2) = b^2 (a + c )^2 (a ^2+b ^2)(b ^2+c ^2) = (ab + bc )^2

13 If b is the meanproportional between a and c, prove that (ab + be) is the meanproportional between (a2 + b2) and (b2 + c2).

Answer»

here it is clearly given that b is the mean proportional between a and c .

therefore, b^2 = ac

Now (a ^2+b^ 2)(b ^2+c ^2) = (a ^2+ ac )( ac +c^ 2) ,

(a ^2+b^ 2)(b ^2+c ^2) = a(a + c ) c( a + c ) ,

(a ^2+b^ 2)(b ^2+c ^2) = a c (a + c )^2 ,

(a ^2+b ^2)(b^ 2+c ^2) = b^2 (a + c )^2

(a ^2+b ^2)(b ^2+c ^2) = (ab + bc )^2