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Given:

Lines x – 3y + 1 = 0 and 2x + 5y – 9 = 0

To find:

The equations of the lines through the point of intersection of the lines x – 3y + 1 = 0 and 2x + 5y – 9 = 0 and whose distance from the origin is.

Explanation:

The equation of the straight line passing through the point of intersection of x 3y + 1 = 0 and 2x + 5y 9 = 0 is given below:

x 3y + 1 + λ(2x + 5y 9) = 0

(1 + 2λ)x + ( 3 + 5λ)y + 1 9λ = 0 … (1)

The distance of this line from the origin is

1 + 81λ2 – 18λ = 145λ2 – 130λ + 50

64λ2 – 112λ + 49 = 0

(8λ – 7)2 = 0

λ

Substituting the value of λ in (1), we get the equation of the required line.

22x + 11y – 55 = 0

2x + y – 5 = 0

Hence, equation of required line is 2x + y – 5 = 0.

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