Find the values of the parameter a so that the point (a, 2) is an interior point of the triangle formed by the lines x + y – 4 = 0, 3x – 7y – 8 = 0 and 4x – y – 31 = 0.

According to ques,:

   

   

and

   

forming a triangle and point (a, 2)is an interior point of the triangle

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

   

   

and

   

respectively.

On solving them, we get A (7, – 3), B (18/5, 2/5) and C (209/25, 61/25) as the coordinates of the vertices.

Let P (a, 2) be the given point.

It is given that point P (a, 2) lies inside the triangle. So, we have the following:

(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.

Thus, if A and P lie on the same side of BC, then

   

   

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

A ∈ (22/3, 33/4)

∴ A ∈ (22/3, 33/4)