Question

If $a,bandcareinA.P$, then $\frac{{(a-c)}^2}{b^2-ac}$is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D

$4$

Explanation for the correct option:

Finding the value:

As $a,bandcareinA.P$

$\therefore b=\frac{a+c}{2}$

Now to find the value of $\frac{{(a-c)}^2}{b^2-ac}$, putting the value of $b$, we get

$$\begin{gathered} \frac{{(a-c)}^2}{b^2-ac}=\frac{{(a-c)}^2}{{\left({\displaystyle \frac{a+c}{2}}\right)}^2-ac} \\ =\frac{4{(a-c)}^2}{{\left({\displaystyle a+c}\right)}^2-4ac} \\ =\frac{4{(a-c)}^2}{a^2+c^2+2ac-4ac} \\ =\frac{4{(a-c)}^2}{a^2+c^2-2ac} \\ =\frac{4{(a-c)}^2}{{(a-c)}^2} \\ =4 \end{gathered}$$

Hence, the correct option is (D).