Find K if (K,2 ) is an interior point of the triangle formed by the lines x+y=4, 3x-7y=8 and 4x-y=31 - Maths - Straight Lines

Question

Find K if (K,2 ) is an interior point of the triangle formed by
the lines x+y=4, 3x-7y=8 and 4x-y=31

Aakash Sharma · Accepted Answer

Let ABC be the triangle then the coordinates of A, B, C are (8.36, 2.44), (7, –3), (3.6, 0.4). The point K (K, 2) will be inside ∆ABC if the following three conditions hold simultaneously.

(i) A and K lie on same side of BC

⇒ (8.36 + 2.44 – 4) (K + 2 – 4) > 0

⇒ 6.8 (K 0 – 2) > 0

⇒ K – 2 > 0

⇒ K > 2 .........(1)

(ii) B and K both lies on same side of AC

⇒ (3 × 7 – 7 (–3) –8) (3K – 7 × 2 – 8) > 0

⇒ 36 (3K – 14 – 8) > 0

⇒ 3K – 22 > 0

$\Rightarrow K > \frac{22}{3}$ .........(2)

(iii) C and K both lies on the same side of AB

⇒ (4 × 3.6 – 0.4 – 31) (4K – 2 – 31) > 0

⇒ –14 (4K – 33) > 0

⇒ 4K – 33 < 0

$\Rightarrow K > \frac{33}{4}$ ........(3)

from (1), (2) and (3) we get

$\frac{22}{3} < K < \frac{33}{4} i.e. K \in [\frac{22}{3}, \frac{33}{4}]$

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Find K if (K,2 ) is an interior point of the triangle formed by the lines x+y=4, 3x-7y=8 and 4x-y=31.