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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