Question -
Answer -
Given:
x – 5y + 6 = 0, x – 3y+ 2 = 0 and x – 2y – 3 = 0 forming a triangle and point P(α2, α)lies inside or on the triangle
Let ABC be thetriangle of sides AB, BC and CA whose equations are x − 5y + 6 = 0,x − 3y + 2 = 0 and x − 2y − 3 = 0, respectively.
On solving theequations, we get A (9, 3), B (4, 2) and C (13, 5) as the coordinates of thevertices.
It is given that pointP (α2, α) lies either inside or on the triangle. The threeconditions are given below.
(i) A and P must lieon the same side of BC.
(ii) B and P must lieon the same side of AC.
(iii) C and P must lieon the same side of AB.
If A and P lie on thesame side of BC, then
(9 – 9 + 2)(α2 –3α + 2) ≥0
(α – 2)(α –1) ≥ 0
α ∈ (- ∞, 1 ] ∪ [ 2, ∞) … (1)
If B and P lie on thesame side of AC, then
(4 – 4 – 3) (α2 –2α – 3) ≥ 0
(α – 3)(α +1) ≤ 0
α ∈ [- 1, 3] … (2)
If C and P lie on thesame side of AB, then
(13 – 25 + 6)(α2 –5α + 6) ≥0
(α – 3)(α –2) ≤ 0
α ∈ [ 2, 3] … (3)
From equations (1),(2) and (3), we get
α∈ [2, 3]
∴ α∈ [2, 3]