Answer :

Given:


x + y – 4 = 0, 3x – 7y – 8 = 0 and 4x – y – 31 = 0 forming a triangle and point (a, 2)is an interior point of the triangle


To find:


Value of a


Explanation:


Let ABC be the triangle of sides AB, BC and CA whose equations are x + y 4 = 0, 3x 7y 8 = 0 and 4x y 31 = 0, respectively.


On solving them, we get A (7, - 3), B and C as the coordinates of the vertices.
Let P (a, 2) be the given point.


Diagram:



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


21 + 21 – 8 – 3a – 14 – 8 > 0


a > … (1)


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



a < … (2)


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




a > 2 … (3)


From (1), (2) and (3), we get:


A


Hence, A


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