Question

Consider an equilateral \( \Delta PQR \), where \( P(3, 5) \) and the side \( QR \) lies on the line \( x + y = 4 \). If the orthocentre of \( \Delta PQR \) is \( (\alpha, \beta) \), then \( 9(\alpha + \beta) \) is equal to:

• A. 46
• B. 48
• C. 50
• D. 52

Hint:

For equilateral triangles, always remember the "Four-in-One" rule: Centroid = Orthocentre = Incentre = Circumcentre. This allows you to use the simple \( 2:1 \) centroid property.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In an equilateral triangle, the orthocentre, circumcentre, and centroid all coincide at the same point. The centroid divides the median (altitude) from the vertex to the opposite side in the ratio \( 2:1 \).

Step 2: Key Formula or Approach:
1. Find the foot of the perpendicular \( M \) from \( P(3, 5) \) to the line \( x + y - 4 = 0 \). 2. The orthocentre \( G(\alpha, \beta) \) divides the segment \( PM \) in the ratio \( 2:1 \).

Step 3: Detailed Explanation:
1. Foot of perpendicular \( M(h, k) \): \( \frac{h-3}{1} = \frac{k-5}{1} = -\frac{(3+5-4)}{1^2+1^2} = -\frac{4}{2} = -2 \). \( h = 3 - 2 = 1 \), \( k = 5 - 2 = 3 \). So \( M = (1, 3) \). 2. Use section formula for \( G(\alpha, \beta) \) with ratio \( 2:1 \) between \( P(3, 5) \) and \( M(1, 3) \): \( \alpha = \frac{2(1) + 1(3)}{2+1} = \frac{5}{3} \). \( \beta = \frac{2(3) + 1(5)}{2+1} = \frac{11}{3} \). 3. Sum \( \alpha + \beta = \frac{5+11}{3} = \frac{16}{3} \). 4. Calculate \( 9(\alpha + \beta) = 9 \times \frac{16}{3} = 3 \times 16 = 48 \).

Step 4: Final Answer:
The value of \( 9(\alpha + \beta) \) is 48.