Question

The combined equation of the lines which pass through the origin and each of which makes an angle of 30° with the line 3x + 2y – 11 = 0 is ______.

Options

• 23x² + 48xy + 3y² = 0

• 23x² + 48xy – 3y² = 0

• 23x² – 48xy + 3y² = 0

• 23x² – 48xy – 3y² = 0

Answer 1

The combined equation of the lines which pass through the origin and each of which makes an angle of 30° with the line 3x + 2y – 11 = 0 is 23x² + 48xy + 3y² = 0.

Explanation:

Slope of the line 3x + 2y – 11 = 0 is `(-3)/2`

Let the slope of required line be m.

∴ tan 30° = `|("m" + 3/2)/(1 - 3/2 "m")|`

⇒ `1/sqrt(3) = |(2"m" + 3)/(2 - 3"m")|`

⇒ 3m² + 48m + 23 = 0  ......(i)

Since, the line passes through origin, its equation is

y = mx

⇒ m = `y/x`

Substituting the value of m in equation (i), we get

`3(y/x)^2 + 48(y/x) + 23` = 0

⇒ 23x² + 48xy + 3y² = 0