Question

Orthocentre of equilateral \(\Delta ABC\) is at the origin. If side \(BC\) lies along \(x + 2\sqrt{2}y = 4\). If coordinates of vertex \(A\) are (a,b). Find the value of \(\lfloor |a + \sqrt{2}b| \rfloor\), where \([ \cdot ]\) denotes G.I.F. :

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

For equilateral triangles centered at the origin, the vertex coordinates are always proportional to the coefficients of the line representing the opposite side.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In an equilateral triangle, the orthocentre, centroid, circumcentre, and incentre all coincide. Thus, the centroid is at the origin \((0,0)\). The distance from the centroid to a vertex is twice the distance from the centroid to the opposite side.
Step 2: Key Formula or Approach:
1. Distance from origin to line \(Lx+My+N=0\) is \(d = \frac{|N|}{\sqrt{L^2+M^2}}\).
2. In equilateral \(\Delta ABC\) with centroid \(G\), \(GA = 2 \times GD\) where \(D\) is the foot of the altitude.
Step 3: Detailed Explanation:
Distance from centroid \(G(0,0)\) to line \(BC\) (\(x + 2\sqrt{2}y - 4 = 0\)):
\[ GD = \frac{|0 + 0 - 4|}{\sqrt{1^2 + (2\sqrt{2})^2}} = \frac{4}{\sqrt{1+8}} = \frac{4}{3} \] Distance from centroid to vertex \(A\):
\[ GA = 2 \times GD = \frac{8}{3} \] The line through \(A\) and \(G\) is perpendicular to \(BC\). Slope of \(BC\) is \(-\frac{1}{2\sqrt{2}}\), so slope of \(AG\) is \(2\sqrt{2}\).
Coordinates of \(A(a,b)\) satisfy \(b = 2\sqrt{2}a\).
Also, \(a^2 + b^2 = (GA)^2\):
\[ a^2 + (2\sqrt{2}a)^2 = (8/3)^2 \implies 9a^2 = 64/9 \implies a^2 = 64/81 \implies a = \pm 8/9 \] Since \(A\) and \(D\) must be on opposite sides of the origin for the centroid to be at the origin:
For \(BC\), \(0 + 0 - 4<0\). Thus for \(A\), \(a + 2\sqrt{2}b - 4>0\).
Substituting \(b = 2\sqrt{2}a \implies a + 8a - 4>0 \implies 9a>4 \implies a = 8/9\).
Then \(b = 16\sqrt{2}/9\).
Evaluate \(|a + \sqrt{2}b|\):
\[ |8/9 + \sqrt{2}(16\sqrt{2}/9)| = |8/9 + 32/9| = |40/9| \approx 4.44 \] Floor value \(\lfloor 4.44 \rfloor = 4\).
Step 4: Final Answer:
The value is 4.