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Question

# Find the equation of circle circumscribing the quadrilateral formed by four lines 3x + 4y âˆ’ 5 = 0, 4x âˆ’ 3y âˆ’ 5 = 0, 3x + 4y + 5 = 0 and 4x âˆ’ 3y + 5 = 0

A

x2 + y2 = 0

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B

x2 + y2 = 2

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C

x2 + y2 = 25

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D

x2 + y2 = 4

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Solution

## The correct option is B x2 + y2 = 2 Given four sides of quadrilateral are L13x + 4y − 5 = 0 L24x − 3y − 5 = 0 L33x + 4y + 5 = 0 L44x − 3y + 5 = 0 Equation of the circle circumscribing a quadrilateral whose sides in order are represented by L1 = 0, L2 = 0, L3 = 0 and L4 = 0 is μL1L3 + λL2L4 = 0 μ(3x + 4y − 5) (3x + 4y + 5) + λ(4x − 3y − 5) (4x − 3y + 5) = 0 μ((3x + 4y)2 − 25) + λ((4x − 3y)2 − 25) = 0 μ (9x2 + 16y2 + 24xy − 25) +λ(16x2 + 9y2 − 24xy − 25) = 0 (9μ + 16λ)x2 + (16μ + 9λ)y2 + (24μ − 24λ)xy − (25μ + 25λ) = 0- - - - - - (1) Since, this is an equation of circle Conditions for circle (1) Coefficient of xy = 0 24μ − 24λ = 0 μ = λ (2) Coefficient of x2 = coefficient of y2 9μ + 16λ = 16μ + 9λ −7μ + 7λ = 0 λ = μ so, given equation of circle is true for any value of λ = μ except λ = μ≠ 0 lets λ = μ = 1 Substituting the value of λ = μ = 1 in equation (1) (9 + 16)x2 + (16 + 9)y2 + 0× xy − (25 + 25) = 0 25x2 + 25y2 − 50 = 0 x2 + y2 = 2 Equation of circle circumscribing the quadrilateral formed by lines 3x + 4y − 5 = 0, 4x − 3y − 5 = 0, 3x + 4y + 5 = 0, 4x − 3y + 5 = 0 is x2 + y2 = 2 Option B is correct

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