Find the values of α so that the point P(α 2, α) lies inside or on the triangle formed by the lines x – 5y + 6 = 0, x – 3y + 2 = 0 and x – 2y – 3 = 0.

According to ques,:

   

   

and

   

forming a triangle and point P(α², α) lies inside or on the triangle

Let ABC be the triangle of sides AB, BC and CA whose equations are:

   

   

and

   

respectively.

On solving the equations, we have A (9, 3), B (4, 2) and C (13, 5) as the coordinates of the vertices.

It is given that point P (α², α) lies either inside or on the triangle. The three conditions are given below.

(i) A and P must lie on the same side of BC.

(ii) B and P must lie on the same side of AC.

(iii) C and P must lie on the same side of AB.If A and P lie on the same side of BC, then

   

   

   

If B and P lie on the same side of AC, then

   

   

   

If C and P lie on the same side of AB, then

   

   

   

From equations (1), (2) and (3), we have