Question

The combined equation of the pair of lines passing through the origin and making an angle \(\frac{π}{4}\) with the line \(3x+y-6 = 0\) is...

• A. \(x^2+3xy-2y^2 = 0\)
• B. \(2x^2-3xy+y^2 = 0\)
• C. \(2x^2-xy+3y^2 = 0\)
• D. \(2x^2+3xy-2y^2 = 0\)

Hint:

Find the slopes of the two lines using the angle formula, then form the product of the two line equations.

Answer 1

The Correct Option is D

Step 1: Direction idea
The given line makes an angle $\alpha$ with the x-axis where $\tan\alpha = -3$. The required lines make angles $\alpha\pm 45^{\circ}$.

Step 2: Slopes
$\tan(\alpha+45^{\circ}) = \frac{-3+1}{1+3} = -\frac12$ and $\tan(\alpha-45^{\circ}) = \frac{-3-1}{1-3} = 2$.

Step 3: Equation
The lines are $x+2y=0$ and $2x-y=0$. Multiplying them gives $(x+2y)(2x-y) = 2x^2+3xy-2y^2 = 0$. Option (D).

Final Answer:
Option D. \[ \boxed{\text{(D)}\ 2x^2+3xy-2y^2=0} \]