Angle between Two Lines

Q.

What is the formula for $\tan 3A$?

Q.

In a triangle $ABC$ if $\tan \left(\frac{A}{2}\right)=\frac{5}{6}$, $\tan \left(\frac{C}{2}\right)=\frac{2}{5}$ then,

1. $a,c$ and $b$ are in $AP$

2. $a,b$ and $c$ are in $GP$

3. $b,a$ and $c$ are in $AP$

4. $a,b$ and $c$ are in $AP$

Q.

A straight line makes an angle of $135°$ with the x-axis and cuts y-axis at a distance $-5$ from the origin. The equation of the line is

1. $2x+y+5=0$

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

3. $x+y+5=0$

4. $x+y+3=0$

Q.

Equation of the bisector of the acute angle between lines $3x-4y+7=0$ and $12x+5y-2=0$ is

1. $21x–77y+100=0$

2. $99x–27y+81=0$

3. $99x+27y+30=0$

4. $21x–77y–100=0$

Q. A straight line $L$ through the point $(3,-2)$ is inclined at an angle ${60}^{\circ }$ to the line $\sqrt{3}x+y=1.$ If $L$ also intersects the $x$-axis, then the equation of line $L$ is

1. $y+\sqrt{3}x+2-3\sqrt{3}=0$
2. $y-\sqrt{3}x+2+3\sqrt{3}=0$
3. $\sqrt{3}y-x+3+2\sqrt{3}=0$
4. $\sqrt{3}y+x-3+2\sqrt{3}=0$

Q. Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point

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

Q.

The angle between the lines xy = 0 is

1. 

2. 

3. 

4. 

Q.

If $\left(\frac{1}{2},\frac{1}{3},n\right)$ are the direction cosines of a line, then the value of $n$ is

1. $\frac{\sqrt{23}}{6}$

2. $\frac{23}{6}$

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

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

Q.

The line joins two points$A(2,0)$, and$B(3,1)$ is rotated about A in an anti-clockwise direction through an angle of $15°$. The equation of the line in the new position is

1. $\sqrt{3}x–y–2\sqrt{3}=0$

2. $x–3\sqrt{y}–2=0$

3. $\sqrt{3}x+y–2\sqrt{3}=0$

4. $x+3\sqrt{y}–2=0$

Q.

Let $PQR$ be a right angled isosceles triangle right angled at $P(2,1)$. If the equation of the line $QR$ is $2x+y=3$, then the equation representing the pair of lines $PQ$ and $PR$ is

1. $3x^2-3y^2+8xy-20x-10y+25=0$

2. $3x^2-3y^2+8xy+20x+10y+25=0$

3. $3x^2-3y^2+8xy+10x+15y+20=0$

4. $3x^2-3y^2-8xy-10x-15y-20=0$

Q. A line cuts off equal intercepts on the co-ordinate axes. The angle made by this line with the positive direction of X-axis is

1. ${135}^{\circ }$
2. ${90}^{\circ }$
3. ${120}^{\circ }$
4. ${45}^{\circ }$

Q.

A line passes through point $(2,2)$ and perpendicular to the line $3x+y=3$. Then, $y-$intercept is

1. $2$

2. $\frac{5}{3}$

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

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

Q.

The slope of a line is double of the slope of another line. If tangents of the angle between them is $\frac{1}{3}$, find the slopes of the other line.

Q.

The coordinates of the point on the curve $y=x^2-3x+2$, where the tangent is perpendicular to the straight line $y=x$ are

1. $(0,2)$

2. $(1,0)$

3. $(-1,6)$

4. $(2,-2)$

Q.

Integrate $\int \frac{x}{x^4+x^2+1}dx$.

Q.

Let $A(3,0,-1),B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the mid point of $AC$. If $G$ divides $BM$ in the in the ratio, $2:1,$ then $\cos \left(\angle GOA\right)$, ($O$ being the origin) is equal to :

1. $\frac{1}{\sqrt{15}}$
2. $\frac{1}{\sqrt{30}}$
3. $\frac{1}{2\sqrt{15}}$
4. $\frac{1}{6\sqrt{10}}$

Q.

If the line $p{x}^2-qxy-y^2=0$ makes the angles $\alpha$ and $\beta$ with $x$ axis, then the value of $\tan \left(\alpha +\beta \right)$ is:

1. $\frac{-q}{1+p}$

2. $\frac{q}{1+p}$

3. $\frac{-q}{1-p}$

4. None of these

Q.

Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axes respectively.

Q.

A line makes angles $\alpha ,\beta ,\gamma$ with the co-ordinate axis. If $\alpha +\beta =90°$, then $\gamma =$

1. 0°

2. 90°

3. 180°

4. None of these

Q.

What is the equation of the bisectors of the angles between the lines represented by $x^2+2xycot\varnothing +y^2=0$?

1. $x^2-y^2=0$

2. $x^2-y^2=xy$

3. $\left(x^2-y^2\right)cot\varnothing =2xy$

4. None of these

Q. The distance of the point (2, 3) from the line 2x – 3y + 9 = 0 measured along a line x – y + 1 =0 is

1. 4
2. 2
3. 
4. 

Q.

The angle between the curves $y=\sin x$ and $y=\cos x$ is

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

2. ${\tan }^{-1}\left(3\sqrt{2}\right)$

3. ${\tan }^{-1}\left(3\sqrt{3}\right)$

4. ${\tan }^{-1}\left(5\sqrt{2}\right)$

Q.

The angle between the lines represented by the equation $a{x}^2+xy+b{y}^2=0$ will be $45°$, if

1. $a=1,b=6$

2. $a=1,b=-6$

3. $a=6,b=1$

4. None of these

Q. Rohit walks $100$ m to the East from point $A,$ then turns right and walks $100$ m, again turns right and walks $20$ m, again turns right and walks $250$ m, and reached point $B$. Then the distance between $A$ and $B$ is

1. $75$ m
2. $180$ m
3. $170$ m
4. $200$ m

Q. The equation of the locus of the point whose distance from the point $P$ and the line $AB$ are equal, is

1. $x^2+9y^2+6xy-54x+62y-241=0$
2. $9x^2+9y^2-6xy-54x-62y-241=0$
3. $x^2+y^2-2x+27x-31y-120=0$
4. $9x^2+y^2-6xy-54x-62y+241=0$

Q.

If the point A is symmetric to the point B(4, –1) with respect to the bisector of the first quadrant, then the length of AB is

1. 

2. 

3. 

4. 

Q. Let the vectors a, b, c and d be such that $(a\times b)\times (c\times d)=0$. Let $P_1$and $P_2$be planes determined by pair of vectors a, b and c, d respectively. Then the angle between $P_1$and $P_2$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{2}$
3. $0^{\circ }$
4. $\frac{\pi }{4}$

Q. A variable line is drawn through origin ${}^{\prime }{O}^{\prime }$. Two points $A$ and $B$ are taken on the line in $1^{\text{st}}$ quadrant such that $OA=1$ unit and $OB=2$ units. Through points $A$ and $B$ two lines are drawn which making an equal angle $\alpha$ with the line $AB$. Then, the locus of point of intersection of these lines is

1. $x^2+y^2=\frac{9+{\tan }^2\alpha }{4}$
2. $x^2-y^2=\frac{9-{\tan }^2\alpha }{4}$
3. $x^2+y^2=\frac{9+{\tan }^2\alpha }{2}$
4. $x^2+y^2=\frac{9+2{\tan }^2\alpha }{4}$

Q.

Find the angle between the lines $x=a$ and $by+c=0$

Q. If $A(-2,1),B(2,3)$ and $C(-2,-4)$ are three points, then the acute angle between the lines $BA$ and $BC$ is

1. ${\tan }^{-1}\left(\frac{2}{3}\right)$
2. ${\tan }^{-1}\left(\frac{3}{2}\right)$
3. ${\tan }^{-1}\left(\frac{1}{3}\right)$
4. ${\tan }^{-1}\left(\frac{4}{3}\right)$