Angle between Pair of Straight Lines

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.

Show that A intersection B is equal to A intersection C need not imply B=C

Q. The acute angle between two lines whose direction cosines are given by the relation between $l+m+n=0and{l}^2+m^2-n^2=0$

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $noneofthese$

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 acute angles between the pairs of lines $3x^2-7xy+4y^2=0$ and $6x^2-5xy+y^2=0$ be ${\theta }_1$ and ${\theta }_2$ respectively, then

1. ${\theta }_1$ = ${\theta }_2$

2. ${\theta }_1$ = 2${\theta }_2$

3. $2{\theta }_1$ = ${\theta }_2$

4. ${\theta }_1$ = 5${\theta }_2$

Q. No of circles touching all the lines x+y-1=0, x-y-1=0, y+1=0 is

Q.

If the equation $y=mx+c$ and $x\cos \left(\alpha \right)+y\sin \left(\alpha \right)=p$ represents the same straight line, then

1. $p=c\sqrt{(1+m^2)}$

2. $c=p\sqrt{(1+m^2)}$

3. $pc=\sqrt{(1+m^2)}$

4. $p^2+c^2+m^2=1$

Q.

Find the acute angle between the lines $2x-y+3=0$ and $x+y+2=0$.

Q.

If $\theta$ is an angle between the lines given by the equation $6x^2+5xy-4y^2+7x+13y-3=0$, then equation of the line passing through the point of intersection of these lines and making an angle $\theta$ with the positive x - axis is

1. $2x-11y+2=0$

2. $11x-2y+13=0$

3. $2x+11y+13=0$

4. $11x+2y-11=0$

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

Q.

For any two setsA and B prove that

A intersection (A' union B ) = A intersection B

Q. The acute angle between the lines $\frac{x-1}{l}=\frac{y+1}{m}=\frac{z}{n}$ and $\frac{x+1}{m}=\frac{y-3}{n}=\frac{z-1}{l},$ where $l>m>n$ and $l,m,n$ are the roots of the cubic equation $x^3+x^2-4x-4=0,$ is

1. ${\cos }^{-1}\frac{1}{9}$
2. ${\cos }^{-1}\frac{2}{9}$
3. ${\cos }^{-1}\frac{1}{3}$
4. ${\cos }^{-1}\frac{4}{9}$

Q. Find the angle between the vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b}$, where
(i) $\overrightarrow{a}=\hat{i}-\hat{j}\mathrm{and}\overrightarrow{b}=\hat{j}+\hat{k}$

(ii) $\overrightarrow{a}=3\hat{i}-2\hat{j}-6\hat{k}\mathrm{and}\overrightarrow{b}=4\hat{i}-\hat{j}+8\hat{k}$

(iii) $\overrightarrow{a}=2\hat{i}-\hat{j}+2\hat{k}\mathrm{and}\overrightarrow{b}=4\hat{i}+4\hat{j}-2\hat{k}$

(iv) $\overrightarrow{a}=2\hat{i}-3\hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}+\hat{j}-2\hat{k}$

(v) $\overrightarrow{a}=\hat{i}+2\hat{j}-\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$

Q. The equation of normal to the hyperbola $3x^2-y^2=1$ having slope $\frac{1}{3}$ is

1. $x-3y=2\sqrt{2}$
2. $x-3y=-2\sqrt{2}$
3. $x-3y=\sqrt{2}$
4. $x-3y=-\sqrt{2}$

Q. If the base $BC$ of a $△ABC$ passes through $(4,8)$ and its other two sides are bisected at right angle by the pair of lines $x^2-9y^2-8xy=0$, then which of the following is (are) CORRECT?

1. The locus of the point $A$ is a circle.
2. The locus of the point $A$ is an ellipse.
3. $\angle BAC=\pi -{\tan }^{-1}\frac{5}{4}$
4. $\angle BAC=\pi -{\tan }^{-1}\frac{4}{5}$

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. Find the cosine of the angle between the vectors $4\hat{i}-3\hat{j}+3\hat{k}\mathrm{and}2\hat{i}-\hat{j}-\hat{k}.$

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.

The solution set of the equation $ta{n}^{-1}x-co{t}^{-1}x=co{s}^{-1}(2-x)$ is

1. [ -1, 0]

2. [ 0, 1 ]

3. [-1, 1 ]

4. [ 1, 3]

Q. If $x^2+2hxy+y^2=0$ represents the equation of the straight lines through origin which makes an angle of $\alpha$ with the straight line $y+x=0$, then

Q. A man of height 6 ft walks at a uniform speed of 9 ft/sec from a lamp fixed at 15 ft height. The length of his shadow is increasing at the rate of
(a) 15 ft/sec
(b) 9 ft/sec
(c) 6 ft/sec
(d) none of these

Q. Find the angle between the planes.
(i) 2x − y + z = 4 and x + y + 2z = 3
(ii) x + y − 2z = 3 and 2x − 2y + z = 5
(iii) x − y + z = 5 and x + 2y + z = 9
(iv) 2x − 3y + 4z = 1 and − x + y = 4
(v) 2x + y − 2z = 5 and 3x − 6y − 2z = 7

Q.

The angle between the lines represented by the equation $\lambda x^2+(1-\lambda {)}^{2}xy-\lambda y^2=0$, is

1. 

2. 

3. 

4. 

Q. The angle between the lines joining the origin to the points of intersection of the line y = 3x + 2 with the curve $x^2$ + 2xy + ${3y}^2$ + 4x + 8y = 11, is

1. 
2. 
3. 
4. 

Q. The angle of intersection of the parabolas y² = 4 ax and x² = 4ay at the origin is

(a) π/6
(b) π/3
(c) π/2
(d) π/4

Q. A man 2 metres high walks at a uniform speed of 6 km/h away from a lamp-post 6 metres high. Find the rate at which the length of his shadow increases.

Q. Prove that the line x+y=3a touches the curve x+y=3axy and also show that the point of contact is ( 1.5a, 1.5a).

Q.

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

1. 

2. 

3. 

4. None of these

Q. Find the angle between the following pairs of lines:
(i) $\overrightarrow{r}=\left(4\hat{i}-\hat{j}\right)+\mathrm{\lambda }\left(\hat{i}+2\hat{j}-2\hat{k}\right)\mathrm{and}\overrightarrow{r}=\hat{i}-\hat{j}+2\hat{k}-\mathrm{\mu }\left(2\hat{i}+4\hat{j}-4\hat{k}\right)$

(ii) $\overrightarrow{r}=\left(3\hat{i}+2\hat{j}-4\hat{k}\right)+\mathrm{\lambda }\left(\hat{i}+2\hat{j}+2\hat{k}\right)\mathrm{and}\overrightarrow{r}=\left(5\hat{j}-2\hat{k}\right)+\mathrm{\mu }\left(3\hat{i}+2\hat{j}+6\hat{k}\right)$

(iii) $\overrightarrow{r}=\mathrm{\lambda }\left(\hat{i}+\hat{j}+2\hat{k}\right)\mathrm{and}\overrightarrow{r}=2\hat{j}+\mathrm{\mu }\left\{\left(\sqrt{3}-1\right)\hat{i}-\left(\sqrt{3}+1\right)\hat{j}+4\hat{k}\right\}$

Q. Write a value of $\int a^x{e}^{x}dx$.