Angle between Pair of Straight Lines

Q.

Acute angle between the lines represented by $(x^2+y^2)\sqrt{3}=4xy$ is

1. $\frac{\pi }{6}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{3}$

4. $\frac{\pi }{12}$

Q.

Tangent of acute angle between pair of straight lines $a{x}^2+2hxy+b{y}^2$ is given by $tan\theta =\left|\frac{2\sqrt{h^2-ab}}{a-b}\right|$

1. True

2. False

Q. The acute angle $\alpha$ between two straight lines having slopes $m_1$ and $m_2$, is given by;
$tan\alpha =\left(\frac{m_1-m_2}{1+m_1{m}_2}\right|$

1. True
2. False

Q. The locus of intersection of the lines $\mathrm{xcos}\alpha +\mathrm{ysin}\alpha =a\mathrm{and}\mathrm{xsin}\alpha -\mathrm{ycos}\alpha =b$ is , where a and b are constants.

1. $x^2-y^2=a^2+b^2$
2. $x^2+y^2=a^2+b^2$
3. $x^2-y^2=a^2-b^2$
4. $x^2+y^2=a^2-b^2$

Q. Let $L_1$ and $L_2$ are two intersecting lines.
If the image of $L_1$ w.r.t. $L_2$ and $L_2$ w.r.t. $L_1$
coincide, then angle between $L_1$ and $L_2$ is-

1. ${35}^{\circ }$
2. ${60}^{\circ }$
3. ${90}^{\circ }$
4. ${45}^{\circ }$

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. $\frac{b}{1+a}$

2. $\frac{-b}{1+a}$

3. $\frac{a}{1+b}$

4. $\frac{a}{1-b}$

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. $\frac{b}{1+a}$

2. $\frac{-b}{1+a}$

3. $\frac{a}{1+b}$

4. $\frac{a}{1-b}$

Q.

If (a + 3b)(3a + b) = $4h^2$, then the angle between the lines represented by $a{x}^2+2hxy+b{y}^2=0$ is

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

2. ${45}^{\circ }$

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

4. ${\tan }^{-1}\frac{1}{2}$

Q. The equation $x^2-5xy+p{y}^2+3x-8y+2=0$ represents a pair of straight lines. If $\theta$ is the acute angle between them, then $\sin \theta$ equals

1. $\frac{1}{\sqrt{50}}$
2. $\frac{1}{7}$
3. $\frac{1}{5}$
4. $\frac{1}{\sqrt{10}}$

Q.

If (a + 3b)(3a + b) = $4h^2$, then the angle between the lines represented by $a{x}^2+2hxy+b{y}^2=0$ is

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

2. ${45}^{\circ }$

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

4. ${\tan }^{-1}\frac{1}{2}$

Q.

The angle between the lines represented by the equation $x^2-2pxy+y^2=0$, is

1. $se{c}^{-1}p$

2. $co{s}^{-1}p$

3. $ta{n}^{-1}p$

4. $ta{n}^{-1}5p$

Q.

Tangent of acute angle between pair of straight lines $a{x}^2+2hxy+b{y}^2$ is given by $tan\theta =\left|\frac{2\sqrt{h^2-ab}}{a-b}\right|$

1. True

2. False

Q.

If the lines $(p-q)x^2+2(p+q)xy+(q-p)y^2=0$ are mutually perpendicular, then

1. p = q

2. q = 0

3. p = 0

4. p and q may have any value

Q. The equation $x^2-5xy+p{y}^2+3x-8y+2=0$ represents a pair of straight lines. If $\theta$ is the acute angle between them, then $\sin \theta$ equals

1. $\frac{1}{\sqrt{50}}$
2. $\frac{1}{7}$
3. $\frac{1}{5}$
4. $\frac{1}{\sqrt{10}}$

Q.

If the lines $(p-q)x^2+2(p+q)xy+(q-p)y^2=0$ are mutually perpendicular, then

1. p = q

2. q = 0

3. p = 0

4. p and q may have any value

Q.

If the angle between the lines represented by the equation $y^2+hxy-x^{2}ta{n}^{2}A=0$ be 2A, then k=

1. 0

2. 1

3. 2

4. tan A

Q.

Acute angle between the lines represented by $(x^2+y^2)\sqrt{3}=4xy$ is

1. $\frac{\pi }{6}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{3}$

4. $\frac{\pi }{12}$

Q. If the angle between two lines is $\frac{\pi }{3}$ and the slope of one of the lines is $\frac{1}{2}$,
then the slope of the other line is/are:

1. $8-5\sqrt{3}$
2. $8+5\sqrt{3}$
3. $5+8\sqrt{3}$
4. $5-8\sqrt{3}$

Q.

If the angle between the lines represented by the equation $y^2+hxy-x^{2}ta{n}^{2}A=0$ be 2A, then k=

1. 0

2. 1

3. 2

4. tan A