Question

The equation of the lines passing through the origin and having slopes $3$ and $-\frac{1}{3}$ is

• A. $3y^2+8xy-3x^2=0$
• B. $3x^2+8xy–3y^2=0$
• C. $3y^2–8xy+3x^2=0$
• D. $3x^2+8xy+3y^2=0$

Answer 1

The correct option is B

$3x^2+8xy–3y^2=0$

Explanation of the correct answer.

Combined equation of line:

We know that the equation of line through origin and slope m is $y=mx.$

When $m=3$, the equation is

$y=3x$

$\Rightarrow$$y–3x=0$

When $m=-\frac{1}{3}$, the equation is

$y=-\frac{1}{3}x$

$\Rightarrow$$3y+x=0$

Now, the combined equation is

$(y–3x)(3y+x)=0$

$\Rightarrow$$3y^2–9xy+xy–3x^2=0$

$\Rightarrow$ $3y^2–8xy–3x^2=0$

$\Rightarrow$ $3x^2+8xy–3y^2=0$

Hence, option $\left(\mathrm{B}\right)$ is the correct answer.