Question

If \[a,b,c\]are in G.P., then
A. \[a\left( {{b^2} + {a^2}} \right) = c\left( {{b^2} + {c^2}} \right)\]
В. \[a\left( {{b^2} + {c^2}} \right) = c\left( {{a^2} + {b^2}} \right)\]
C. \[{a^2}(b + c) = {c^2}(a + b)\]
D. None of these

Answer 1

Hint:
First, we'll write the conditional equation for variables a, b, and c. Next, we'll note the value of b and c in terms of ‘a’ and the common ratio to create a relation in \[a,{\rm{ }}b\] and \[c\]. Finally, we'll use those equations to find the needed answer. We know that when \[a,{\rm{ }}b\] and \[c\]are in general principle, then \[{b^2} = ac\], which means that when we prove \[{b^2} = ac\], then it is obvious that \[a,{\rm{ }}b\] and \[c\] are in general principle, then we must use the data provided in question to prove this.
Formula use:
\[a,{\rm{ }}b\] and \[c\] are in G.P.
 \[{b^2} = ac\]
Complete step-by-step solution
Given the data in the question that,\[a,{\rm{ }}b\] and \[c\] are in G.P.
We can infer that the common ratio will remain the same from the given data, which is that items a, b, and c are in G.P.
\[\frac{{\rm{b}}}{{\rm{a}}} = \frac{{\rm{c}}}{{\rm{b}}} \ldots \ldots \ldots \ldots {\rm{ (i) }}\]
\[ \Rightarrow {{\rm{b}}^2} = {\rm{ac}} \ldots \ldots \ldots {\rm{ (ii) }}\]
Let ‘r’ be the common ratio:
\[\therefore {\rm{b}} = {\rm{ar}}\]
On squaring both sides of the above equation, we obtain
\[{{\rm{b}}^2} = {({\rm{ar}})^2}\]
That implies,
\[ \Rightarrow {{\rm{b}}^2} = {{\rm{a}}^2}{{\rm{r}}^2}\]
Now let’s calculate for the give data:
\[{b^2}(a - c) = ac(a - c)\]
We have to multiply the term outside to the terms inside the parentheses in the above equation, we obtain
\[ \Rightarrow {b^2}a - {b^2}c = {a^2}c - a{c^2}\]
Now, we have to group similar terms in the above equation, according to the given question:
\[ \Rightarrow a\left( {{b^2} + {c^2}} \right) = c\left( {{a^2} + {b^2}} \right)\]
Therefore, if \[a,b,c\] are in G.P., then \[a\left( {{b^2} + {c^2}} \right) = c\left( {{a^2} + {b^2}} \right)\]
Therefore, the correct option is B.

Note:
Due to the complexity of the calculations and equation, students should exercise caution when carrying out each step. Signs in these questions are subject to student error. Always consider what identity we can use to prove the requested information whenever you are faced with situations of this nature. By continuing in this manner, your solution will be proven correct.