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in Mathematics by (53.3k points)

Determine all values of α for which the point (α, α2) lies inside the triangle formed by the lines 2x + 3y − 1 = 0, x + 2y − 3 = 0 and 5x − 6y − 1 = 0.

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Let L ≡ 2x + 3y − 1 = 0,La' ≡  + 2y − 3 = 0 and L"  5x − 6y − 1 = 0. Vertices are A (5/4, 7/8), B (1/3, 1/9) and C (−7, 5). See Fig.

1 .  For the side BC (L = 0), the points A(5/4, 7/8) and (α, α2) must be on the same side

 Therefore,

α < -1 or α > 1/3    ...(1)

 2.   For the side CA (L' = 0), the points B(1/3, 1/9) and (αα2) are on the same side. Therefore

 3. For the side AB (L" = 0), the points C and (αα2) are on the same side. Therefore

 From Eqs. (1) - (3) ,α ∈ (-3/2, -1) ∪ (1/2, 1).

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