With usual notations, in any $\triangle A B C$, if a $\cos B=b \cos A$, then the triangle

1. an isosceles triangle
2. an equilateral triangle
3. a right angled triangle
4. a scalene triangle

Solution

We know that $\frac{a}{\sin A}=\frac{b}{\sin B}$ From (1) and (2), we write $\frac{\mathrm{a}}{\mathrm{b}}=\frac{\sin \mathrm{A}}{\sin \mathrm{B}}=\frac{\cos \mathrm{A}}{\cos \mathrm{B}} \Rightarrow \tan \mathrm{A}=\tan \mathrm{B} \Rightarrow \mathrm{A}=\mathrm{B}$ Thus $\Delta$ is an isosceles triangle.

Asked in: MHT CET 2021 (22 Sep Shift 2)

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