Pair of Lines Passing Though Origin

Q.

If the lines represented by the equation $a{x}^2-bxy-y^2=0$ make angles $\alpha$ and $\beta$ with the x - axis, then $\tan (\alpha +\beta )$ =

1. 

2. None of these

3. 

4. 

Q.

Equation $6x^2-5xy+y^2=0$ represents____________________

1. Pair of coincident lines

2. Pair of imaginary lines with real point of intersection i.e.(0, 0)

3. Pair of distinct straight lines

4. None of these

Q.

If the lines represented by the equation $2x^2-3xy+y^2=0$ make angles $\alpha$ and $\beta$ with x - axis, then $co{t}^2\alpha +co{t}^2\beta =$

1. 0

2. $\frac{3}{2}$

3. $\frac{7}{4}$

4. $\frac{5}{4}$

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-1, b-2, c-3

2. a-2, b-1, c-3

3. a-3, b-2, c-1

4. a-3, b-1, c-2

Q. The line $x=c$ cuts the triangle with corners $(0,0);(1,1)$ and $(9,1)$ into two regions. For the area of the two regions to be the same $c$ must be equal to

1. $512$
2. $3$
3. $3$ or $15$
4. $7/2$