analytical geomentry(pair of straight lines)?

Question

find the angle between the pair of lines 6x(square)+13xy+6y(square)+8x+7y+2=0

Answer 1

Find the angle between the pair of lines given by the equation:

6x² + 13xy + 6y² + 8x + 7y + 2 = 0

This equation is that of a conic section that degenerates into a pair of intersecting lines.

Solve for y in terms of x.

6y² + (13x + 7)y + (6x² + 8x + 2) = 0

Factoring we get:

(2y + 3x + 1)(3y + 2x + 2) = 0

y = (-3x - 1)/2
y = (-2x - 2)/3

Rewrite the equations of the lines.

y = (-3/2)x - 1/2
y = (-2/3)x - 2/3

The slopes of the lines are the tangent of the angle of the line from the positive x-axis.

tan(α) = -3/2
tan(β) = -2/3

arctan(-3/2) = α
arctan(-2/3) = β

Use the angle subtraction formula for tangents.

tan(α - β) = tan[arctan(-3/2) - arctan(-2/3)]

= [(-3/2) - (-2/3)] / [1 + (-3/2)(-2/3)]

= (-5/6) / [1 + 1] = (-5/6)/2 = -5/12

arctan(-5/12) ≈ -22.6°

Since the direction doesn't matter, the angle between the two lines is approximately 22.6°.