Question

Assertion :If $u$ and $v$ are two unit vectors inclined at an angle $\alpha$ and $x$ is a unit vector bisecting the angle between them then $x=\frac{u+v}{2\cos \frac{\alpha }{2}}$. Reason: If $ABC$ be an isosceles triangle with $AB=AC=1$ then vector representing bisector of angle $A$ is Given by $AD=\frac{AB+AC}{2}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$\begin{array}{c}Assertion\\ Bi\sec torx=\frac{\overrightarrow{u}+\overrightarrow{v}}{\left|\overrightarrow{u}+\overrightarrow{v}\right|}\\ =\frac{\overrightarrow{u}+\overrightarrow{v}}{\sqrt{\overrightarrow{u}+\overrightarrow{v}+2uv\cos \alpha }}\\ =\frac{\overrightarrow{u}+\overrightarrow{v}}{\sqrt{2+2\cos \alpha }}\\ =\frac{\overrightarrow{u}+\overrightarrow{v}}{2\cos \frac{\alpha }{2}}\\ Reason\\ \left|x\right|=AB\cos \frac{\alpha }{2}\\ =\cos \frac{\alpha }{2}\\ \hat{x}=\frac{AB.AC}{2\cos \frac{\alpha }{2}}\\ \therefore x=AD=\cos \frac{\alpha }{2}.\frac{AB.AC}{2\cos \frac{\alpha }{2}}\\ =\frac{AB+AC}{2}\end{array}$

We find that the option $A$ is correct and complete.