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JEE Main 2026 · 5 Apr · Shift 1, Mathematics question 11

In an equilateral triangle PQR, let the vertex P be at (3,5)(3,5) and the side QR be along the line x+y=4x+y=4. If the orthocentre of the triangle PQR is (α,β)(\alpha,\beta), then 9(α+β)9(\alpha+\beta) is equal to:

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Answer and worked solution

Official answer: D

Orthocentre. In an equilateral triangle it coincides with the centroid, which lies on the altitude from P, two-thirds of the way to QR.

Foot of the altitude. From P(3,5)P(3,5) to x+y=4x+y=4: 3+5−42=2\dfrac{3+5-4}{2}=2, so the foot is M=(1,3)M=(1,3).

Centroid. P+23(M−P)=(53,113)P+\tfrac23(M-P)=\left(\tfrac53,\tfrac{11}3\right), so α+β=163\alpha+\beta=\tfrac{16}3 and 9(α+β)=489(\alpha+\beta)=48.

(D)

Topics:Distance of a point from a lineDistance and section formulae
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