Pair of Lines

Q. The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is

1. 
2. 
3. 
4. 

Q. If the pair of straight lines $a{x}^2+2hxy+b{y}^2=0$ is rotated about the origin through ${90}^{\circ },$ then the equations in the new position is

1. $a{x}^2-2hxy+b{y}^2=0$
2. $a{x}^2+2hxy-b{y}^2=0$
3. $b{x}^2+2hxy+a{y}^2=0$
4. $b{x}^2-2hxy+a{y}^2=0$

Q.

The lines represented by the equation $9x^2+24xy+16y^2+21x+28y+6=0$ are

1. Parallel

2. Coincident

3. Perpendicular

4. skew lines

Q. Let a, b, c and d be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies in the fourth quadrant and is equdistant from the two axes, then

1. $2bc-3d=0$
2. $2bc+3ad=0$
3. $2ad-3bc=0$
4. $3bc+2ad=0$

Q. The lines joining the origin to the point of intersection of $3x^2+mxy-4x+1=0$ and $2x+y-1=0$ are at right angles. Then all possible values of $m$ lie in the interval

1. $R$
2. $[1,2]$
3. $\varphi$
4. $(1,2]$

Q. The distance between the two lines represented by the equation $9x^2-24xy+16y^2-12x+16y-12=0$ is __ units

1. $\frac{8}{5}$
2. $\frac{6}{5}$
3. $\frac{11}{5}$
4. $\frac{9}{5}$

Q. If the equation of the lines passing through point $(1,1)$, one making an angle $\theta$ with the positive direction of $x-$axis and the other making the same angle with the positive direction of $y-$axis, is $x^2-(a+2)xy+y^2+a(x+y-1)=0,a\ne -2,$ then the value of sin $2\theta$ is

1. $a-2$
2. $a+2$
3. $\frac{2}{(a+2)}$
4. $\frac{2}{a}$

Q. The equation of one of the lines represented by the pair of lines $12x^2-10xy+2y^2+11x-5y+2=0$ is/are

1. $6x-2y+1=0$
2. $3x-y+2=0$
3. $2x-y+2=0$
4. $4x-2y+1=0$

Q.

The lines $(lx+my{)}^2-3(mx-ly{)}^2=0$ and lx + my + n = 0 form

1. An isosceles triangle

2. A right angled triangle

3. An equilateral triangle

4. concurent lines

Q. Number of points with integral co-ordinates that lie inside a triangle whose co-ordinate are (0, 0), (0, 21) and (21, 0)

1. 210
2. 190
3. 220
4. none of these

Q. The straight lines represented by $(y-mx{)}^2=a^2(1+m^2)$ and $(y-nx{)}^2=a^2(1+n^2)$ forms
(where $mn\ne -1$ and $m\ne n$)

1. square
2. rhombus
3. rectangle
4. trapezium

Q. If the equation of the lines passing through point $(1,1)$, one making an angle $\theta$ with the positive direction of $x-$axis and the other making the same angle with the positive direction of $y-$axis, is $x^2-(a+2)xy+y^2+a(x+y-1)=0,a\ne -2,$ then the value of sin $2\theta$ is

1. $a-2$
2. $a+2$
3. $\frac{2}{(a+2)}$
4. $\frac{2}{a}$