JEE Main · 2026 · 05 Apr 2026 · Shift 1 · Straight Lines

In an equilateral triangle PQR , let the vertex P be at

(3, 5)

and the side QR be along the line

x + y = 4

. If the orthocentre of the triangle

P QR

(α, β)

, then

9 (α + β)

is equal to :

(A)16
(B)27
(C)36
(D)48✓ Correct

Answer: (D)

Solution

The altitude from

P (3, 5)

to the base

QR : x + y = 4

meets it at its midpoint

M

, found from

\frac{x - 3}{1} = \frac{y - 5}{1} = - \frac{3 + 5 - 4}{1 ^{2} + 1 ^{2}} = - 2

, giving

M = (1, 3)

. In an equilateral triangle the orthocentre coincides with the centroid, which divides the median

P M

from the vertex in the ratio

2 : 1

, so it is

P + \frac{2}{3} (M - P) = (\frac{5}{3}, \frac{11}{3})

. Hence

9 (α + β) = 9 (\frac{5}{3} + \frac{11}{3}) = 9 \cdot \frac{16}{3} = 48

JEE Main · 2026 · 05 Apr 2026 · Shift 1 · Single correct option

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