Let a triangle be bounded by the lines L 1 : 2 x + 5 y = 10 ; L 2 : - 4 x + 3 y = 12 and the line L 3 , which passes through the point P 2 , 3 , intersect L 2 at A and L 1 at B . If the point P divides the line-segment A B , internally in the ratio 1 : 3 , then the area of the triangle is equal to

Step-by-step solution

Given, point A lies on L 2   :   - 4 x + 3 y = 12 Take x = α ,   so   y = 4 + 4 3 α , A α , 4 + 4 3 α Points B lies on L 1   :   2 x + 5 y = 10 Take x = β ,   so   y = 2 - 2 5 β , B β , 2 - 2 5 β Now point P divides A B internally in the ratio 1 : 3 ⇒ P 2 , 3 = P 3 α + β 4 , 3 4 + 4 3 α + 1 2 - 2 5 β 4 ⇒ α = 3 13 , β = 95 13 We get, point A 3 13 , 56 13 , B 95 13 , - 12 13 Vertex C of triangle i