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Third vertex of a triangle from two vertices and its orthocentre

This type of question, asked 3 times: 6 Apr 2024, 9 Apr 2024, 22 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    6 Apr 2024

    165
    If P(6,1)P(6,1) be the orthocentre of the triangle whose vertices are A(5,−2),B(8,3)A(5, - 2),B(8,3) and C(h,k)C(h,k), then the point CC lies on the circle.

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  2. Question 2

    9 Apr 2024

    156
    Two vertices of a triangle ABCABC are A(3,−1)A(3, - 1) and B(−2,3)B( - 2,3), and its orthocentre is P(1,1)P(1,1). If the coordinates of the point CC are (α,β)(\alpha,\beta) and the centre of the circle circumscribing the triangle PABPAB is (h,k)(h,k), then the value of (α+β)+2( h+k)(\alpha + \beta) + 2(\text{ }h + k) equals :

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  3. Question 3

    22 Jan 2026 S2

    57
    Among the statements (S1) : If A(5,−1)A(5, - 1) and B(−2,3)B( - 2,3) are two vertices of a triangle, whose orthocentre is (0,0)(0,0), then its third vertex is (−4,−7)( - 4, - 7) and (S2) : If positive numbers 2a,b,c2a,b,c are three consecutive terms of an A.P., then the lines ax+by+c=0ax+by+c=0 are concurrent at (2,−2)(2,-2).

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