Find the equation of the straight line passing through the point of in

1.	Find the equation of the straight line passing through the point of intersection of 2x + 3y + 1 = 0 and 3x – 5y – 5 = 0 and equally inclined to the axes.
Answer» Given: The equations, 2x + 3y + 1 = 0 and 3x – 5y – 5 = 0 The equation of the straight line passing through the points of intersection of 2x + 3y + 1 = 0 and 3x − 5y − 5 = 0 is 2x + 3y + 1 + λ(3x − 5y − 5) = 0 (2 + 3λ)x + (3 − 5λ)y + 1 − 5λ = 0 y = – [(2 + 3λ) / (3 – 5λ)] – [(1 – 5λ) / (3 – 5λ)] The required line is equally inclined to the axes. So, the slope of the required line is either 1 or − 1. So, – [(2 + 3λ) / (3 – 5λ)] = 1 and – [(2 + 3λ) / (3 – 5λ)] = -1 -2 – 3λ = 3 – 5λ and 2 + 3λ = 3 – 5λ λ = 5/2 and 1/8 Now, substitute the values of λ in (2 + 3λ)x + (3 − 5λ)y + 1 − 5λ = 0, we get the equations of the required lines as: (2 + 15/2)x + (3 – 25/2)y + 1 – 25/2 = 0 and (2 + 3/8)x + (3 – 5/8)y + 1 – 5/8 = 0 19x – 19y – 23 = 0 and 19x + 19y + 3 = 0 ∴ The required equation is 19x – 19y – 23 = 0 and 19x + 19y + 3 = 0

Find the equation of the straight line passing through the point of intersection of 2x + 3y + 1 = 0 and 3x – 5y – 5 = 0 and equally inclined to the axes.

Answer»

Given:

The equations, 2x + 3y + 1 = 0 and 3x – 5y – 5 = 0

The equation of the straight line passing through the points of intersection of 2x + 3y + 1 = 0 and 3x − 5y − 5 = 0 is

2x + 3y + 1 + λ(3x − 5y − 5) = 0

(2 + 3λ)x + (3 − 5λ)y + 1 − 5λ = 0

y = – [(2 + 3λ) / (3 – 5λ)] – [(1 – 5λ) / (3 – 5λ)]

The required line is equally inclined to the axes. So, the slope of the required line is either 1 or − 1.

So,

– [(2 + 3λ) / (3 – 5λ)] = 1 and – [(2 + 3λ) / (3 – 5λ)] = -1

-2 – 3λ = 3 – 5λ and 2 + 3λ = 3 – 5λ

λ = 5/2 and 1/8

Now, substitute the values of λ in (2 + 3λ)x + (3 − 5λ)y + 1 − 5λ = 0, we get the equations of the required lines as:

(2 + 15/2)x + (3 – 25/2)y + 1 – 25/2 = 0 and (2 + 3/8)x + (3 – 5/8)y + 1 – 5/8 = 0