Pair of Lines

Q.

If the sum of the slopes of the lines represented by the equation $x^2-2xy\tan A-y^2=0$ be $4$, then $\angle A=$

1. $0°$

2. $45°$

3. $60°$

4. ${\tan }^{-1}\left(-2\right)$

Q.

If one of the lines represented by the equation $a{x}^2+2hxy+b{y}^2=0$ be $y=mx$, then

1. $b{m}^2+2hm+a=0$

2. $b{m}^2+2hm–a=0$

3. $a{m}^2+2hm+b=0$

4. $a{m}^2-2hm+b=0$

Q.

Two lines represented by the equation $x^2+xy+y^2=0$ are

1. Coincident

2. Parallel

3. Mutually perpendicular

4. Imaginary

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.

Find $m$ if the line $y=mx-2$ intersects $y=x^2$ in just one point.

Q. The area enclosed by the quadrilateral formed by $x^2{y}^2-9x^2-25y^2+225=0$ is

1. $60$ sq.units
2. $100$ sq.units
3. $30$ sq.units
4. $20$ sq.units

Q. The equation of line passing through the point of intersection of the lines4x-3y-1=0 and 5x-2y-3=0 and parallel to the line 2y-3x+2=0, is

1. x-3y=1
2. 3x-2y=1
3. 2x-3y=1
4. 2x-y=1

Q. The straight lines given by x(a+2b)+y(a+3b)=a+b for different values of a and b passes through a fixed point. The coordinates of the fixed point are

1. (2, -1)
2. (2, 1)
3. (-2, 1)
4. (-2, -1)

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. If one of the lines of $m{y}^2+(1-m^2)xy-m{x}^2=0$ is a bisector of the angle between the lines $xy=0$, then $m$ can be

1. $1$
2. $2$
3. $-1$
4. $-\frac{1}{2}$

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. Equation of a line which is parallel to the line common to the pair of lines given by $6x^2-xy-12y^2=0$ and $15x^2+14xy-8y^2=0$ and at a distance of $7$ units from it, is

1. $3x+4y=35$
2. $3x+4y+35=0$
3. $2x-3y=35$
4. $2x-3y+35=0$

Q. The length of the perpendicular(s) from the origin to the line passing through $(2,2)$ and perpendicular to the lines$x^2+2xy-3y^2=0$, is/are

1. $\frac{2\sqrt{2}}{\sqrt{5}}$ units
2. $\frac{2}{\sqrt{5}}$ units
3. $2\sqrt{2}$ units
4. $2$ units

Q. The area enclosed by the quadrilateral formed by $x^2{y}^2-9x^2-25y^2+225=0$ is

1. $60$ sq.units
2. $100$ sq.units
3. $30$ sq.units
4. $20$ sq.units

Q. The combined equation of two sides of a triangle is $x^2-3y^2-2xy+8y-4=0.$ The third side, which is variable always passes through the point $(-5,-1)$. If the range of values of the slope of the third line such that the origin is an interior point of the triangle is $(a,b)$, then the value of $(a+\frac{1}{b})$ is

1. $\frac{1}{2}$
2. $2$
3. $0$
4. $4$

Q.

Angles made by the lines represented by the equation xy + y = 0 with y-axis are

1. $0^{\circ }$ and ${90}^{\circ }$

2. $0^{\circ }$ and ${30}^{\circ }$

3. ${30}^{\circ }$ and ${60}^{\circ }$

4. ${30}^{\circ }$ and ${90}^{\circ }$

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 acute angle between the pair of straight lines passing through $(-6,-8)$ and also through the points which divide the line $2x+y+10=0$ enclosed between coordinate axes in the ratio $1:2:2$ in the direction from the point of intersection with the $x-$axis to the point of intersection with $y-$axis is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{12}$

Q. If the adjacent sides of a parallelogram are represented by $2x^2-5xy+3y^2=0$ and the equation of one diagonal is $x+y-2=0,$ then the equation of the other diagonal is

1. $9x-11y=0$
2. $9x+11y=0$
3. $11x-9y=0$
4. $11x+9y=0$

Q. Product of perpendicular distances drawn from origin to pair of straight lines $12x^2+25xy+12y^2+10x+11y+2=0$ is __ sq. units

1. $\frac{1}{25}$
2. $\frac{2}{25}$
3. $\frac{3}{25}$
4. $\frac{4}{25}$