Condition for Two Lines to Be Perpendicular

Q.

The angle between the lines $2x-y+3=0$ and $x+2y+3=0$ is

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

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

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

4. ${60}^{\circ }$

Q. Let the equations of two sides of a triangle be $3x-2y+6=0$ and $4x+5y-20=0$. If the orthocentre of this triangle is at $(1,1)$, then the equation of its third side is :

1. $26x+61y+1675=0$
2. $122y+26x+1675=0$
3. $122y-26x-1675=0$
4. $26x-122y-1675=0$

Q.

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

1. $\frac{1}{3}$

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

3. $1$

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

Q.

The set of all possible values of $\theta$in the interval $(0,\pi )$ for which the points$(1,2)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y=1$is

1. $\left(0,\frac{\mathrm{\pi }}{4}\right)$

2. $\left(0,\frac{\pi }{2}\right)$

3. $\left(0,\frac{3\mathrm{\pi }}{4}\right)$

4. $\left(\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right)$

Q. The value of k such that $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}$ lies in the plane 2x -4y +z =7, is

1. 7
2. -7
3. No real value
4. 4

Q.

Name The Type Of Quadrilateral Formed By The Following Points - $(4,5),(7,6),(4,3),(1,2)$ And Give Reasons For Your Answer

Q.

Line $L$ has intercepts $a$ and $b$ on the coordinate axes. When the axes are rotated through a fixed angle keeping the origin fixed, the same line $L$ has intercepts $p$ and $q$, then

1. $a^2+b^2=p^2+q^2$

2. $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}+\frac{1}{q^2}$

3. $a^2+p^2=b^2+q^2$

4. $\frac{1}{a^2}+\frac{1}{p^2}=\frac{1}{b^2}+\frac{1}{q^2}$

Q.

If $a,b,c$ are in AP, then the straight line $ax+by+c=0$will always pass through the point

1. $(-1,-2)$

2. $(1,-2)$

3. $(-1,2)$

4. $(1,2)$

Q.

If a line with y-intercept $2$, is perpendicular to the line $3x-2y=6$, then its x-intercept is

1. $1$

2. $2$

3. $-4$

4. $3$

Q. Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2y-z=3.$ If $d$ is the distance of the point $(0,0,6)$ from $L,$ then $d^2$ is equal to

Q.

The equation of the base of an equilateral triangle is $x+y=2$and the vertex is $(2,-1)$. The length of the side of the triangle is

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

2. $\sqrt{2}$

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

4. None of these

Q.

The mirror image of the point $(1,2,3)$ in a plane is $\left(\frac{-7}{3},\frac{-4}{3},\frac{-1}{3}\right)$. Which of the following points lies on this plane

1. $(1,-1,1)$

2. $(-1,-1,1)$

3. $(1,1,1)$

4. $(-1,-1,-1)$

Q.

The straight lines whose direction cosines are given by al + bm + cn = 0, fmn + gnl + hlm = 0 are perpendicular, if

1. 
   

2. 
   

3. 
   

4. 
   

Q. For real numbers $\alpha$ and $\beta \ne 0$, if the point of intersection of the straight lines $\frac{x-\alpha }{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and $\frac{x-4}{\beta }=\frac{y-6}{3}=\frac{z-7}{3}$, lies on the plane $x+2y-z=8,$ then $\alpha –\beta$ is equal to

1. $9$
2. $5$
3. $3$
4. $7$

Q.

If $2p$ is the length of the perpendicular from the origin to the lines $\frac{x}{a}+\frac{y}{b}=1$, then $a^2,8p^2,b^2$ are in

1. AP

2. GP

3. HP

4. None of these

Q.

The image of the point $(1,3,4)$ with respect to the plane $2\mathrm{x}-\mathrm{y}+\mathrm{z}+3=0$ is

1. $\left(-1,4,3\right)$

2. $\left(-3,5,2\right)$

3. $\left(1,3,4\right)$

4. $\left(-1,-3,-4\right)$

Q. If the two lines $x+(a-1)y=1$ and $2x+a^{2}y=1,(a\in R-\{0,1\left\}\right)$ are perpendicular, then the distance of their point of intersection from the origin is :

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

Q. Two vertices of a triangle are $(0,2)$ and $(4,3)$. If its orthocentre is at the origin, then its third vertex lies in which quadrant?

1. first
2. third
3. fourth
4. second

Q. If the straight line, $2x-3y+17=0$ is perpendicular to the line passing through the points $(7,17)$ and $(15,\beta )$ then $\beta$ equals:

1. $-5$
2. $\frac{35}{3}$
3. $5$
4. $-\frac{35}{3}$

Q.

The equation of pair of lines joining the origin to the point of intersection of $x^2+y^2=9$ and $x+y=3$ is:

1. ${(x^2+-x)}^2=9$

2. $xy=0$

3. ${(3+y)}^2+y^2=9$

4. ${(x-y)}^2=9$

Q. The equation of the line through the point $(0,1,2)$ and perpendicular to the line
$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :

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

Q. The slope of any line which is perpendicular to the $x$-axis is .......... .

1. $1$
2. $-1$
3. $0$
4. Not defined

Q. The angle between the straight lines, whose direction cosines are given by the equations $2l+2m-n=0$ and $mn+nl+lm=0,$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{2}$
3. $\pi -{\cos }^{-1}\left(\frac{4}{9}\right)$
4. ${\cos }^{-1}\left(\frac{8}{9}\right)$

Q.

The equation $x^2+k_1{y}^2+k_{2}xy=0$ represents a pair of perpendicular lines, if

1. $k_1=-1$

2. $k_1=2k_2$

3. $2k_1=k_2$

4. None of these

Q.

Find the equation of the straight line passing through (-2, 3) and inclined at an angle of ${45}^{\circ }$ with the x-axis.

Q.

If a line lies in the octant $OXYZ$ and it makes equal angles with the axis, then

1. $l=m=n=\frac{1}{\surd 3}$

2. $l=m=n=\pm \frac{1}{\surd 3}$

3. $l=m=n=-\frac{1}{\surd 3}$

4. $l=m=n=\pm \frac{1}{\surd 2}$

Q.

Find the equation of the straight line which passes through the point (1, 2) and makes such an angle with the positive direction of x-axis whose sine is $\frac{3}{5}.$

Q.

The equation of the line passing through $(1,5)$ and perpendicular to the line $3x-5y+7=0$ is

1. $5x+3y-20=0$

2. $3x-5y+7=0$

3. $3x-5y+6=0$

4. $5x+3y+7=0$

Q. A line $y=x+2$ is drawn on the co-ordinate plane. This line is rotated by ${90}^{\circ }$ clockwise about the point $(0,2).$ Another line $y=-2x+10$ is drawn such that a triangle is formed by these three lines. If the area of the triangle is $A$ sq. unit, then the value of $3A$ is

Q. If the straight lines $2x+3y-3=0$ and $x+ky+7=0$ are perpendicular, then the value of $k$ is

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