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Centroid from the circumcentre and orthocentre (Euler line, ratio 2:1)

This type of question, asked 2 times: 8 Apr 2024, 22 Jan 2025. All 2 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    8 Apr 2024

    82
    If the ortho centre of the triangle formed by the lines 2x+3y−1=0,x+2y−1=02x+ 3y- 1 = 0,x+ 2y- 1 = 0 and ax+by−1=0ax + by - 1 = 0, is the centroid of another triangle, whose circumcentre and orthocentre respectively are (3,4)(3,4) and (−6,−8)( - 6, - 8), then the value of ∣a−b∣|a - b| is

  2. Question 2

    22 Jan 2025

    147
    Let A(6,8),B(10cos⁡α,−10sin⁡α)A(6,8),B(10\cos\alpha, - 10\sin\alpha) and C (−10sin⁡α,10cos⁡α)( - 10\sin\alpha,10\cos\alpha), be the vertices of a triangle. If L(a,9)L(a,9) and G(h,k)G(h,k) be its orthocenter and centroid respectively, then (5a−3h+6k+100sin⁡2α)(5a - 3h + 6k + 100\sin2\alpha) is equal to ______\_\_\_\_\_\_