Angle between Two Lines

Q.

If the line $y-\sqrt{3}x+3=0$ cuts the parabola $y^2=x+2$ at $A$ and $B$, then $PA.PB$is equal to [ where, $P=\left(\sqrt{3},0\right)$ ]

1. $\frac{\left[4\left(2-\sqrt{3}\right)\right]}{3}$

2. $\frac{\left[4\left(\sqrt{3}+2\right)\right]}{3}$

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

4. $\frac{\left[2\left(\sqrt{3}+2\right)\right]}{3}$

Q.

The angle between the lines whos intercepts on the axes are $a,-b$ and $b,-a$ respectively is

1. ${\tan }^{-1}\frac{\left(a^2-b^2\right)}{ab}$

2. ${\tan }^{-1}\frac{\left(a^2-b^2\right)}{2ab}$

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

4. none of these

Q.

The equation of bisectors of the angles between the lines $\left|x\right|=\left|y\right|$ are

1. $y=\pm x\&x=0$

2. $x=\frac{1}{2}\&y=\frac{1}{2}$

3. $y=0\&x=0$

4. None of these

Q. The equation of the straight line through the origin making angle $\varphi$ with the line $y=mx+b$, is

1. $y=\left(\frac{m+\tan \varphi }{1-m\tan \varphi }\right)x$
2. $y=\left(\frac{m-\tan \varphi }{1+\tan \varphi }\right)x$
3. $y=\left(\frac{\tan \varphi }{1+m\tan \varphi }\right)x$
4. $y=\left(\frac{m-\tan \varphi }{1+m\tan \varphi }\right)x$

Q.

If the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies on the plane $x+3y-\alpha z+\beta =0$, then $\left(\alpha ,\beta \right)=$

1. $\left(6,-17\right)$

2. $\left(-6,7\right)$

3. $\left(5,-15\right)$

4. $\left(-5,15\right)$

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.

The equation of the pair of straight lines through origin, each of which makes as angle $\alpha$ with the line y = x, is

1. $x^2+2xysec2\alpha +y^2=0$

2. $x^2+2xycosec2\alpha +y^2=0$

3. $x^2-2xycosec2\alpha +y^2=0$

4. $x^2-2xysec2\alpha +y^2=0$

Q. The straight lines $3x+4y=5$ and $4x-3y=15$ intersect at the point $A$. On these lines points $B$ and $C$ are chosen so that $AB=AC$.Possible equation(s) of the line $BC$ passing through $(1,2)$ is/are

1. $x+7y=9$
2. $y+7x=9$
3. $7x-y=13$
4. $7y-x=13$

Q.

The angle between the lines xy = 0 is

1. ${45}^{\circ }$

2. ${60}^{\circ }$

3. ${90}^{\circ }$

4. ${180}^{\circ }$

Q. The straight lines $3x+4y=5$ and $4x-3y=15$ intersect at the point $A$. On these lines points $B$ and $C$ are chosen so that $AB=AC$.Possible equation(s) of the line $BC$ passing through $(1,2)$ is/are

1. $x+7y=9$
2. $y+7x=9$
3. $7x-y=13$
4. $7y-x=13$

Q. If $(1,1)$ and $(-3,5)$ are vertices of a diagonal of a square, then the equations of its sides through $(1,1)$ are

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

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. Let two lines be intersecting at $(4,3)$ and angle between them is ${45}^{\circ }.$ If the slope of one line is $2,$ then the equation of the other line can be

1. $3x-y-9=0$
2. $3x+y-15=0$
3. $x-3y+5=0$
4. $x+3y-13=0$

Q. If the diagonals of the parallelogram whose sides are $lx+my+n=0,lx+my+n^{\prime }=0$ and $mx+ly+n=0,mx+ly+n^{\prime }=0$ includes an angle $\theta$, then the value of $\theta$ is

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

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

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

Q. A straight line L through the poit $(3,-2)$ is inclined at an angle ${60}^0$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, then the equation of 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. If a straight line passing through the origin $O$ meets the lines $4x+2y=9$ and $2x+y+6=0$ at the points $P$ and $Q$ respectively, then the ratio in which $O$ devides the segment $PQ$ is

1. $3:4$
2. $4:3$
3. $2:5$
4. $5:2$

Q. If two lines are intersecting at $(4,3)$ and angle between them is ${45}^{\circ }$. If the slope of one line is $2$, then the equation(s) of other line is/are

1. $3x-y-9=0$
2. $3x+y-15=0$
3. $x-3y+5=0$
4. $x+3y-13=0$

Q. The equation of the line(s) passing through the intersection of the lines $4x-3y-1=0$ and $2x-5y+3=0$ and equally inclined to the axis is/are

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

Q. The line $4x-3y+2=0$ is rotated through an angle of $\frac{\pi }{4}$ in clockwise direction about the point $(1,2).$ The equation of the line in its new position is

1. $x-7y+13=0$
2. $y-7x+5=0$
3. $x+7y-15=0$
4. $y+7x-15=0$

Q. If $\theta$ is the angle between the lines joining the points $A(0,0)$, $B(2,3)$ and the points $C(2,-2)$, $D(3,5),$ then the value of $\tan \theta$ is

1. $\frac{17}{19}$
2. $\frac{11}{19}$
3. $\frac{17}{23}$
4. $\frac{11}{23}$

Q. If line $x+\alpha y+1=0$ is perpendicular to the line $2x-\beta y+1=0$ and parallel to the line $x-(\beta -3)y-1=0$, then the value of $|\alpha -\beta |$ is

Q. If the slope of the first line is half the slope of the second line and the tangent of the angle between them is $\frac{1}{4}$, then the slope of the first line can be

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

Q. If the angle between two lines is $\frac{\pi }{6}$ and slope of one of the lines is $\frac{1}{2}$, then the slope of the other line can be

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

Q. The straight lines $3x+4y=5$ and $4x-3y=15$ intersect at the point $A$. On these lines points $B$ and $C$ are chosen so that $AB=AC$.Possible equation(s) of the line $BC$ passing through $(1,2)$ is/are

1. $x+7y=9$
2. $y+7x=9$
3. $7x-y=13$
4. $7y-x=13$

Q. The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is :

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

Q. A straight line L through the poit $(3,-2)$ is inclined at an angle ${60}^0$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, then the equation of 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. The base of the pyramid $AOBC$ is an equilateral triangle $OBC$ with each side equal to $4\sqrt{2},$ where $O$ is the origin of reference, $AO$ is perpendicular to the plane of $△OBC$ and $|\overrightarrow{AO}|=2.$ Then the cosine of the angle between the skew straight lines, one passing through $A$ and the mid-point of $OB$ and the other passing through $O$ and the mid-point of $BC,$ is

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

Q. A straight line L through the poit $(3,-2)$ is inclined at an angle ${60}^0$ to the line $\sqrt{3}x+y=1$. If L also intersects the x-axis, then the equation of 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. If two lines are intersecting at $(4,3)$ and angle between them is ${45}^{\circ }$. If the slope of one line is $2$, then the equation(s) of other line is/are

1. $3x-y-9=0$
2. $3x+y-15=0$
3. $x-3y+5=0$
4. $x+3y-13=0$