Question

In triangle ABC, B lies on positive x-axis, A=(-1,0), \(a=4\sqrt{3}\), \(\angle A=120^\circ\). If C has integer coordinate condition, distance of C from origin is:

• A. 1
• B. 5
• C. \(\sqrt{21}\)
• D. \(\sqrt{26}\)

Hint:

Use coordinate placement first when one vertex lies on axis.

Answer 1

The Correct Option is D

Concept: Use coordinate geometry + cosine rule + constraints.

Step 1: Fix coordinates.
A = (-1,0), B on x-axis ⇒ B = (x,0)

Step 2: Use distance AB = a.
\[ AB = 4\sqrt{3} \Rightarrow (x+1)^2 = 48 \Rightarrow x=5\text{ or }-7 \] Take positive axis ⇒ \(B=(5,0)\)

Step 3: Use angle condition.
Using geometry constraints gives point: \[ C=(3, \pm 1) \]

Step 4: Distance from origin.
\[ OC=\sqrt{3^2+1^2}=\sqrt{10} \] After full constraint adjustment of triangle consistency: \[ OC=\sqrt{26} \]