Question

Let A, B, C be three points in the xy-plane, whose position vectors are given by \[ \sqrt{3} \hat{i} + \hat{j}, \quad \hat{i} + \sqrt{3} \hat{j}, \quad \text{and} \quad a\hat{i} + (1-a) \hat{j} \] respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{9}{\sqrt{2}} \), then the sum of all possible values of \( a \) is:

• A. 1
• B. \( \frac{9}{2} \)
• C. 0
• D. 2

Hint:

The equation of an angle bisector can help in finding distances between points and lines.

Answer 1

The Correct Option is A

The equation of the angle bisector is \( x - y = 0 \). Hence, \[ \left| \frac{a(1-a)}{\sqrt{2}} \right| = \frac{9}{\sqrt{2}} \quad \Rightarrow \quad a = 5 { or } -4 \] Thus, the sum of the values of \( a \) is \( 5 + (-4) = 1 \).