Question

Let \( ABC \) be an equilateral triangle with orthocenter at the origin and the side \( BC \) lying on the line \( x+2\sqrt{2}\,y=4 \). If the coordinates of the vertex \( A \) are \( (\alpha,\beta) \), then the greatest integer less than or equal to \( |\alpha+\sqrt{2}\beta| \) is:

• A. \(2\)
• B. \(4\)
• C. \(5\)
• D. \(3\)

Hint:

In equilateral triangle coordinate problems, placing the centroid or orthocenter at the origin greatly simplifies calculations using symmetry.

Answer 1

The Correct Option is D

Concept: In an equilateral triangle, the orthocenter, centroid, and circumcenter coincide. Hence, if the orthocenter is at the origin, the centroid of the triangle is also at the origin. Further, the centroid divides each median in the ratio \(2:1\).
Step 1: Use the centroid condition Let the coordinates of vertices be: \[ A(\alpha,\beta),\quad B(x_1,y_1),\quad C(x_2,y_2) \] Since the centroid is at the origin: \[ \alpha+x_1+x_2=0,\quad \beta+y_1+y_2=0 \quad \cdots (1) \]
Step 2: Use the fact that \( BC \) lies on the given line The midpoint of \( BC \) is: \[ \left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right) \] Using (1): \[ \text{Midpoint of } BC=\left(-\frac{\alpha}{2},-\frac{\beta}{2}\right) \] Since this midpoint lies on the line \( x+2\sqrt{2}\,y=4 \): \[ -\frac{\alpha}{2}+2\sqrt{2}\left(-\frac{\beta}{2}\right)=4 \] \[ \Rightarrow -\alpha-2\sqrt{2}\beta=8 \] \[ \Rightarrow \alpha+2\sqrt{2}\beta=-8 \quad \cdots (2) \]
Step 3: Find the required expression We need \( |\alpha+\sqrt{2}\beta| \). From (2), \[ \alpha+2\sqrt{2}\beta=-8 \] Divide into two parts: \[ \alpha+\sqrt{2}\beta = -8 - \sqrt{2}\beta \] To maximize \( |\alpha+\sqrt{2}\beta| \), use the fact that the altitude of an equilateral triangle passes through the centroid and is perpendicular to the base \( BC \). The direction vector of \( BC \) is \( (2\sqrt{2},-1) \), so the direction of altitude is \( (1,2\sqrt{2}) \). Hence, \[ (\alpha,\beta)=t(1,2\sqrt{2}) \] Substitute into (2): \[ t+2\sqrt{2}(2\sqrt{2}t)=t+8t=9t=-8 \Rightarrow t=-\frac{8}{9} \]
Step 4: Compute the required value \[ \alpha+\sqrt{2}\beta =t(1+4)=5t=-\frac{40}{9} \] \[ |\alpha+\sqrt{2}\beta|=\frac{40}{9}\approx 4.44 \] The greatest integer \( \le 4.44 \) is: \[ \boxed{3} \]