# 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.

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 Rate this question :

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