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 .

Question

Philoid · Accepted Answer

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.

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.