Question

the combined equation of the lines passing through the origin and having slopes 3 and -2 is

Answer 1

Since the lines are passing through the origin then (0,0) will be the points of both ,

equation of first line

y - 0 = 3(x-0)

y = 3x ----(1)

equation of second line ,

y = -2x -----(2)

combining(1) & (2 We get (y+2x)(2y-x) = 0)
We get 6x² - y² + xy = 0

Answer 2

equation of a line passing through origin(0,0) with slope 3 i
(y-0) = 3(x-0)
⇒ y = 3x 
⇒ y - 3x = 0 ---------(1)

equation of a line passing through origin(0,0) with slope -2 is
(y-0) = -2(x-0)
⇒ y = -2x 
⇒ y + 2x = 0 ---------(2)

Combining lines (1) and (2), we get
(y - 3x)(y + 2x) = 0
⇒ y² + (2x-3x)y - 6x² = 0
⇒ y² - 6x² -xy = 0
⇒ 6x² - y² + xy = 0

Equation of combination of lines is 6x² - y² + xy = 0