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Question

Find the equation of the circle circumscribing the triangle formed by the lines x + y = 0, 2x + y = 4 and x + 2y = 5


A

x2 + y2 + 17x + 19y + 50 = 0

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B

x2 + y2 + 17x - 11y + 30 = 0

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C

x2 + y2 - 17x - 11y + 30 = 0

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D

x2 + y2 - 17x - 19y + 50 = 0

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Solution

The correct option is D

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+35y2515×575y+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


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