Pair of Lines Passing Though Origin

Q.

Separate equations of lines, for a pair of lines, whose equation is $x^2+xy-12y^2=0$ are

1. $x+4y=0$ and $x+3y=0$

2. $2x-3y=0$ and $x-4y=0$

3. $x-6y=0$ and $x-3y=0$

4. $x+4y=0$ and $x-3y=0$

Q.

If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common then the joint equation of the other two lines is given by

1. $3x^2+8xy-3y^2=0$

2. $3x^2+10xy+3y^2=0$

3. $y^2+2xy-3x^2=0$

4. $x^2+2xy-3y^2=0$

Q.

$a{x}^2+2hxy+b{y}^2=0$ always represents a pair of straight lines passing through the origin. If
$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ \text{a.}{h}^2>ab& \text{1. Lines are coincident}\\ \text{b.}{h}^2=ab& \text{2. Lines are real and distinct}\\ \text{c.}{h}^2<ab\text{3. Lines are imaginary with real point of intersection i.e. (0, 0)}& \end{array}$

1. a-2 b-1 c-3

2. a-3 b-2 c-1

3. a-3 b-1 c-2

4. a-1 b-2 c-3

Q. Prove that the equation
$y^3-x^3+3xy(y-x)=0$
represents three straight lines equally inclined to one another.