If the angle between the pair of lines x^2+2 2 x y+k y^2=0, k >0 is 45^ , then the area (in square units) of the triangle formed by the pair of bisectors of the angles between these lines and the line x+2 y+1=0 is

Step-by-step solution

Given: x^2+2 2 x y+k y^2=0 ...(i) x+2 y+1=0 ...(ii) The angle between the lines (i) is 45^ . 45^ = | 1+k (1-k)+8 | 1 2 = | 1+k 9-k | aligned & 1 2 = (1+k)^2 (9-k) 2(1+k)^2=(9-k) & 2 (1+k^2+2 k )=9-k & 2 k^2+5 k-7=0 & 2 k^2+7 k-2 k-7=0 & (2 k+7)(k-1)=0 k= -7 2 , 1 aligned But k>0 k=1 The equation of angular bisector of (i) is aligned & h (x^2-y^2 )-(a-b) x y=0 & 2 (x^2-y^2 )-(1-k)=0 aligned ...(3) The required area formed by the lines (ii), (iii) is n^2 h^2-a b |a m^2-2 h l m+b l^2 | = 1 3 .