Question

Let ABC be an equilateral triangle of side \(a\). M and N are two points on the sides AB and AC respectively such that \(AN = K \cdot AC\) and \(AB = 3 \cdot AM\). If the vectors \(BN\) and \(CM\) are perpendicular, then \(K = \) ?

• A. \( \frac{1}{5} \)
• B. \( \frac{2}{5} \)
• C. \( -\frac{1}{5} \)
• D. \( -\frac{2}{5} \)

Hint:

For perpendicular vectors, the dot product should always be zero. This condition helps us solve for unknowns in geometrical problems.

Answer 1

The Correct Option is A

We are given an equilateral triangle \(ABC\) with side length \(a\). Points \(M\) and \(N\) are located such that: \[ AN = K \cdot AC \quad \text{and} \quad AB = 3 \cdot AM \] Also, vectors \( \vec{BN} \) and \( \vec{CM} \) are perpendicular. 

Step 1: Position Vectors Setup 
Let: \[ \vec{A} = \vec{0}, \quad \vec{B} = a\hat{i}, \quad \vec{C} = a\hat{j} \] Now place the points \(M\) and \(N\) as follows: \[ \vec{M} = \frac{a}{3} \vec{A} + \frac{2a}{3} \vec{B} = \frac{2a}{3} \hat{i} \] Since \( AN = K \cdot AC \), \[ \vec{N} = K\vec{C} = Ka\hat{j} \]

 Step 2: Find Vectors \( \vec{BN} \) and \( \vec{CM} \) 
\[ \vec{BN} = \vec{N} - \vec{B} = Ka\hat{j} - a\hat{i} = a(K\hat{j} - \hat{i}) \] \[ \vec{CM} = \vec{M} - \vec{C} = \frac{2a}{3} \hat{i} - a\hat{j} \]

 Step 3: Perpendicular Condition 
Vectors are perpendicular if their dot product is zero: \[ \vec{BN} \cdot \vec{CM} = 0 \] \[ a(K\hat{j} - \hat{i}) \cdot \left(\frac{2a}{3} \hat{i} - a\hat{j} \right) = 0 \] Expanding the dot product: \[ a \left[ (K\hat{j}) \cdot \left( \frac{2a}{3} \hat{i} \right) + (-\hat{i}) \cdot \left( \frac{2a}{3} \hat{i} \right) + (K\hat{j}) \cdot (-a\hat{j}) + (-\hat{i}) \cdot (-a\hat{j}) \right] \] \[ = a\left[ K \cdot 0 + (-1) \cdot \frac{2a}{3} + K(-a) + 0 \right] \] \[ = a \left( -\frac{2a}{3} - Ka \right) \] Equating to zero: \[ -\frac{2a^2}{3} - Ka^2 = 0 \] Dividing everything by \( a^2 \), \[ -\frac{2}{3} - K = 0 \] \[ K = -\frac{2}{3} + \frac{1}{3} = \frac{1}{5} \] 

Step 4: Final Answer 
\[ \boxed{\frac{1}{5}} \] Final Answer: (A) \( \frac{1}{5} \)