Search any question & find its solution
Question:
Answered & Verified by Expert
In $\triangle \mathrm{ABC}$, if a, b, c are in arithmetic progression and $\mathrm{C}=2 \mathrm{~A}$, then $\mathrm{a}: \mathrm{c}=$
Options:
Solution:
1519 Upvotes
Verified Answer
The correct answer is:
$2: 3$
Given that $a, b, c$ are in A.P
$\begin{aligned}
& \Rightarrow \mathrm{b}=\frac{\mathrm{a}+\mathrm{c}}{2} \ldots \text { (i) } \\
& \text { And } \mathrm{C}=2 \mathrm{~A} \Rightarrow \sin \mathrm{C}=\sin 2 \mathrm{~A}=2 \sin \mathrm{A} \cdot \cos \mathrm{A} \\
& \Rightarrow \mathrm{C}=2 \mathrm{a} \cdot \frac{\left(\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2\right)}{2 \mathrm{bc}} \\
& \Rightarrow 2 \mathrm{c}^3+3 \mathrm{a}^3-3 \mathrm{ac}^2-2 \mathrm{a}^2 \mathrm{c}=0 \\
& \Rightarrow 3\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^3-2\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^2-3\left(\frac{\mathrm{a}}{\mathrm{c}}\right)+2=0 \\
& \Rightarrow\left(3\left(\frac{\mathrm{a}}{\mathrm{c}}\right)-2\right)\left[\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^2-1\right]=0 \Rightarrow \frac{\mathrm{a}}{\mathrm{c}}=\frac{2}{3}=2: 3
\end{aligned}$
$\begin{aligned}
& \Rightarrow \mathrm{b}=\frac{\mathrm{a}+\mathrm{c}}{2} \ldots \text { (i) } \\
& \text { And } \mathrm{C}=2 \mathrm{~A} \Rightarrow \sin \mathrm{C}=\sin 2 \mathrm{~A}=2 \sin \mathrm{A} \cdot \cos \mathrm{A} \\
& \Rightarrow \mathrm{C}=2 \mathrm{a} \cdot \frac{\left(\mathrm{b}^2+\mathrm{c}^2-\mathrm{a}^2\right)}{2 \mathrm{bc}} \\
& \Rightarrow 2 \mathrm{c}^3+3 \mathrm{a}^3-3 \mathrm{ac}^2-2 \mathrm{a}^2 \mathrm{c}=0 \\
& \Rightarrow 3\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^3-2\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^2-3\left(\frac{\mathrm{a}}{\mathrm{c}}\right)+2=0 \\
& \Rightarrow\left(3\left(\frac{\mathrm{a}}{\mathrm{c}}\right)-2\right)\left[\left(\frac{\mathrm{a}}{\mathrm{c}}\right)^2-1\right]=0 \Rightarrow \frac{\mathrm{a}}{\mathrm{c}}=\frac{2}{3}=2: 3
\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.