Find the equation of the circle circumscribing the triangle formed by the lines x + y = 0, 2x + y = 4 and x + 2y = 5
x2 + y2 − 17x − 19y + 50 = 0
Family of circles circumscribing a triangle whose sides are given by L1 = 0, L2 = 0 and L3 = 0 is
L1L2 + λL2L3 + μL3L1 = 0
(x + y − 6) (2x + y − 4) + λ(2x + y − 4) (x + 2y − 5) + μ (x + 2y − 5) (x + y − 6)
(2x2 + xy − 4x + 2xy + y2 − 4y − 12x − 6y + 24)
+ λ(2x2 + 4xy − 10x + xy + 2y2 − 5y − 4x − 8y + 20)
+ μ(x2 +xy − 6x + 2y2 − 12y − 5x − 5y + 30)
= (2 + 2λ+ μ)x2 + (1 + 2λ + 2μ)y2+ (3 + 5λ + 3μ)xy + (−16 − 14λ − 11μ)x+ (−10 − 13λ − 17μ)y + (24 + 20λ + 30μ) = 0- - - - - - (1)
Since, it is an equation of circle
So coefficient of xy should be zero and coefficient of x2 = coefficient of y2
Coefficient of xy = zero
So, 3 + 5λ + 3μ = 0
5λ + 3μ = −3- - - - - - (2)
Coefficient of x2 = coefficient of y2
2 + 2λ + μ = 1 + 2λ + 2μ
μ= 1
Substituting μ in equation 2
5λ+ 3 = −3
λ=−65
Now, substituting the values of λ&μ in equation (1)
(2+2(−65)+1)x2+(1+2(−65)+2×1)y2+0 xy
−16+14(−65)+11×1)x−(10+13(−65)+17×1)y+(24+20(−65)+30×1)
35x2+35y2−515×575y+30=0
Multiplying both sides by 53
x2 + y2 − 17x − 19y + 50 = 0
so, equation of circle circumscribing triangle is
x2 + y2 − 17x − 19y + 50 = 0