Question

If a,b,c are in AP , then \[\dfrac{{{{\left( {a - c} \right)}^2}}}{{\left( {{b^2} - ac} \right)}} = \]
1. \[1\]
2. \[2\]
3. \[3\]
4. \[4\]

Answer 1

Hint: We are given three terms that are in AP. For solving this we first need to know what Arithmetic Progression (AP) is .A mathematical sequence in which the difference between two consecutive terms is always a constant and it is abbreviated as AP. The fixed number that must be added to any term of an AP to get the next term is known as the common difference of the AP.

Complete step-by-step solution:
Following are the properties of an Arithmetic Progression:
1. If the same number is added or subtracted from each term of an A.P, then the resulting terms in the sequence are also in A.P with the same common difference.
2. If each term in an A.P is divided or multiply with the same non-zero number, then the resulting sequence is also in an A.P
3. Three numbers \[a,b,c\] are in AP then \[2b = a + c\].
4. If we select terms in the regular interval from an A.P, these selected terms will also be in AP.
Since we are given \[a,b,c\] are in AP
Therefore we have the expression \[2b = a + c\]
Or we can say that \[b = \dfrac{{a + c}}{2}\]
Now consider the expression \[\dfrac{{{{\left( {a - c} \right)}^2}}}{{\left( {{b^2} - ac} \right)}} = \dfrac{{{{\left( {a - c} \right)}^2}}}{{{{\left( {\dfrac{{a + c}}{2}} \right)}^2} - ac}}\]
\[ = \dfrac{{4{{\left( {a - c} \right)}^2}}}{{{a^2} + 2ac + {c^2} - 4ac}}\]
On simplification we get the following expression,
\[\dfrac{{{{\left( {a - c} \right)}^2}}}{{\left( {{b^2} - ac} \right)}}\] \[ = \dfrac{{4{{\left( {a - c} \right)}^2}}}{{{{\left( {a - c} \right)}^2}}} = 4\]
Therefore option (4) is the correct answer.

Note: To solve such type of questions one must have a strong grip over the concept of Arithmetic Progression (AP). In addition to this we must know the basic properties of Arithmetic Progression so that we can simplify the terms at each step. Do the calculations carefully and recheck them so as to get the required correct answer and to minimize the errors.