Condition for Two Lines to Be Perpendicular

Q.

A line L is perpendicular to the line $5x-y=1$ and the area of the triangle formed by line $L$and coordinate axes is $5$. The equation of the line $L$ is

1. $x+5y=5$

2. $x+5y=\pm 5\surd 2$

3. $x–5y=5$

4. $x–5y=\pm 5\surd 2$

Q.

From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that $\angle QPR$ is a right angle, then the possible value(s) of
$\lambda$ is (are)

1. $\sqrt{2}$

2. 1

3. -1

4. $-\sqrt{2}$

Q. If the vertices of a triangles are $A(7,-1),B(-2,8)$ and $C(1,2)$ then which of following is/are correct?

1. Altitude through $A$ is $x-2y-9=0$
2. Altitude through $B$ is $2x-y+12=0$
3. Altitude through $C$ is $x-y+1=0$
4. Orthocentre is $(2,4)$

Q. Let $A(6,4)$ and $B(2,12)$ be two given points, then the equation of perpendicular bisector of $AB$ is

1. $x+2y-12=0$
2. $x-2y-12=0$
3. $x+2y+12=0$
4. $x-2y+12=0$

Q.

From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that $\angle QPR$ is a right angle, then the possible value(s) of
$\lambda$ is (are)

1. $\sqrt{2}$

2. 1

3. -1

4. $-\sqrt{2}$

Q. If the slope of a line is $K$ and it make an angle of $\frac{\pi }{4}$ with another line which passess through $(K,0)$ and $(8,5)$ then the value(s) of $K$ is/are

1. $6+\sqrt{39}$
2. $6-\sqrt{39}$
3. $12-\sqrt{39}$
4. $12+\sqrt{39}$

Q. Let $l_1$ be parallel to $\frac{x}{5/3}+\frac{y}{5/4}=1$ and $l_2$ be perpendicular to it. If $l_1$ has $x-$intercept equals to $2$ units and $l_2$ has $y-$intercept equals to $2$ units, then the absolute value of the ratio of $y-$intercept of $l_1$ to $x-$intercept of $l_2$ is

Q. The equation of the line which is perpendicular to $x+4y-5=0$ at it's $y$ intercept

1. $16x+4y-5=0$
2. $16x-4y-5=0$
3. $16x+4y+5=0$
4. $16x-4y+5=0$

Q. If the two lines $(a+b)x-4y+5=0$ and $x+(a-b)y+10=0$ are perpendicular to each other then

1. $a=b$
2. $2a+b=0$
3. $3a-5b=0$
4. $5a-2b=0$

Q. If the foot of the perpendicular drawn from the point $(1,0,3)$ on a line passing through $(\alpha ,7,1)$ is $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$, then $\alpha$ is equal to

Q. Direction cosines of the line which is perpendicular to the lines whose direction ratios are 1, –1, 2 and 2, 1, –1 are given by :

1. $[\frac{-1}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{3}{\sqrt{35}}]$
2. $[\frac{-1}{\sqrt{35}},\frac{-5}{\sqrt{35}},\frac{3}{\sqrt{35}}]$
3. $[\frac{1}{\sqrt{35}},\frac{5}{\sqrt{35}},\frac{-3}{\sqrt{35}}]$
4. $[\frac{1}{\sqrt{35}},\frac{-7}{\sqrt{35}},\frac{-3}{\sqrt{35}}]$

Q. If $l_1,m_1,n_1$ and $l_2,m_2,n_2$ are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be

1. $(m_1{n}_2-m_2{n}_1),(n_1{l}_2-l_1{n}_2),(l_1{m}_2-l_2{m}_1)$
2. $(l_1{l}_2-m_2{m}_1),(m_1{m}_2-n_1{n}_2),(n_1{n}_2-l_2{l}_1)$
3. $\frac{1}{\sqrt{l_1^2+m_1^2+n_1^2}},\frac{1}{\sqrt{l_2^2+m_2^2+n_2^2}},\frac{1}{\sqrt{3}}$
4. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Q. If the vertices of a triangles are $A(7,-1),B(-2,8)$ and $C(1,2)$ then which of following is/are correct?

1. Altitude through $A$ is $x-2y-9=0$
2. Altitude through $B$ is $2x-y+12=0$
3. Altitude through $C$ is $x-y+1=0$
4. Orthocentre is $(2,4)$

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. second
3. third
4. fourth

Q.

A line l passing through the origin is perpendicular to the lines
$l_1:(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k},-\infty <t<\infty l_2:(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k},-\infty <s<\infty$
Then, the coordinate(s) of the point(s) on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of l and $l_1$ is (are)

1. $(\frac{7}{3},\frac{7}{3},\frac{5}{3})$

2. $-1,-1,0$

3. $(1,1,1)$

4. $(\frac{7}{9},\frac{7}{9},\frac{8}{9})$

Q. If $l_1,m_1,n_1$ and $l_2,m_2,n_2$ are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be

1. $(m_1{n}_2-m_2{n}_1),(n_1{l}_2-l_1{n}_2),(l_1{m}_2-l_2{m}_1)$
2. $(l_1{l}_2-m_2{m}_1),(m_1{m}_2-n_1{n}_2),(n_1{n}_2-l_2{l}_1)$
3. $\frac{1}{\sqrt{l_1^2+m_1^2+n_1^2}},\frac{1}{\sqrt{l_2^2+m_2^2+n_2^2}},\frac{1}{\sqrt{3}}$
4. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$

Q. A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2).$ If $O$ is the origin and $P$ is $(2,-1,1),$ then the projection of $\overrightarrow{OP}$ on this plane is of length :

1. $\sqrt{\frac{2}{3}}$
2. $\sqrt{\frac{2}{11}}$
3. $\sqrt{\frac{2}{5}}$
4. $\sqrt{\frac{2}{7}}$

Q. The equation of the perpendicular bisector of the line segment joining $(2,1)$ and $(3,4)$ is

1. $x+3y-10=0$
2. $x+3y+10=0$
3. $x-3y-10=0$
4. $x-y+10=0$

Q. $\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{a. Lines}\frac{x-1}{-2}=\frac{y+2}{3}=\frac{z}{-1}\text{and}& \\ \overrightarrow{r}=(3\hat{i}-\hat{j}+\hat{k})+t(\hat{i}+\hat{j}+\hat{k})\text{are}& \text{p. intersecting}\\ \text{b. Lines}\frac{x+5}{1}=\frac{y-3}{7}=\frac{z+3}{3}\text{and}& \\ x-y+2z-4=0=2x+y-3z+5=0& \\ \text{are}& \text{q. perpendicular}\\ \text{c. Lines}(x=t-3,y=-2t+1,z=-3t-2)& \\ \text{and}\overrightarrow{r}=(t+1)\hat{i}+(2t+3)\hat{j}+(-t-9)\hat{k}& \\ \text{are}& \text{r. parallel}\\ \text{d. Lines}\overrightarrow{r}=(\hat{i}+3\hat{j}-\hat{k})+t(2\hat{i}-\hat{j}-\hat{k})\text{and}& \\ \overrightarrow{r}=(-\hat{i}-2\hat{j}+5\hat{k})+s(\hat{i}-2\hat{j}+\frac{3}{4}\hat{k})\text{are}& \text{s. skew}\end{array}$

Then which of the following is correct ?

1. $a\to q,s;b\to r;c\to p,q;d\to p$
2. $a\to q;b\to r;c\to p,q;d\to p,q$
3. $a\to s;b\to r,s;c\to p;d\to p,q$
4. $a\to q,s;b\to r,s;c\to p;d\to p$

Q. Equation of a line in the plane $\pi :2x-y+z-4=0$ which is perpendicular to the line $l$ whose equation is $\frac{x-2}{1}=\frac{y-2}{-1}=\frac{z-3}{-2}$ and which passes through the point of intersection of $l$ and $\pi$ is

1. $\frac{x-2}{1}=\frac{y-1}{5}=\frac{z-1}{-1}$
2. $\frac{x-1}{3}=\frac{y-3}{5}=\frac{z-5}{-1}$
3. $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z+1}{1}$
4. $\frac{x-2}{2}=\frac{y-1}{-1}=\frac{z-1}{1}$

Q. The equation of the straight line which passes through the intersection of the lines $x+2y=5$ and $3x+7y=17$ and is perpendicular to the line $3x+4y=14$ is

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

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 the line which is perpendicular to $x+4y-5=0$ at it's $y$ intercept

1. $16x+4y-5=0$
2. $16x-4y-5=0$
3. $16x+4y+5=0$
4. $16x-4y+5=0$

Q. The line joining (2, -5) and
(-2, 5) is perpendicular to the line joining (6, 3) and (1, 1).

1. True
2. False

Q. If the lines $x=ay+b,z=cy+d$ and $x=a^{\prime }z+b^{\prime },y=c^{\prime }z+d^{\prime }$ are perpendicular, then :

1. $a{b}^{\prime }+b{c}^{\prime }+1=0$
2. $b{b}^{\prime }+c{c}^{\prime }+1=0$
3. $a{a}^{\prime }+c+c^{\prime }=0$
4. $c{c}^{\prime }+a+a^{\prime }=0$

Q. Let a square with side length ${}^{\prime }{p}^{\prime }$ and making an angle of $\theta$ with $x-$ axis, has one vertex at origin. If $0<\theta <\frac{\pi }{2}$, then the equation of the diagonals of the square is

1. $x(1+\tan \theta )-y(1+\tan \theta )=py(1+\tan \theta )=x(1-\tan \theta )$
2. $x(\cos \theta +\sin \theta )-y(\cos \theta +\sin \theta )=py(\cos \theta +\sin \theta )=x(\cos \theta +\sin \theta )$
3. $x(\cos \theta -\sin \theta )+y(cos\theta +\sin \theta )=py(\cos \theta -\sin \theta )=x(\cos \theta +\sin \theta )$
4. $x(1+\tan (\frac{\pi }{4}+\theta )-y(1+\tan (\frac{\pi }{4}+\theta )=py\left(\tan \right(\frac{\pi }{4}+\theta \left)\right)=x\left(\tan \right(\frac{\pi }{4}-\theta \left)\right)$

Q. The value of x for which the line through A(-2, 6) and B(4, 8) is perpendicular to the the line through the points
C(8, 12) and D(x, 24) is .

1. 4
2. 1
3. 6
4. 8

Q. If in a $\Delta ABC,A\equiv (1,10),$ circumcentre $\equiv (-\frac{1}{3},\frac{2}{3})$ and orthocentre $\equiv (-\frac{11}{3},\frac{4}{3}),$ then the equation of side $BC$ is

1. $12x+39y+155=0$
2. $12x-39y+155=0$
3. $12x+39y-155=0$
4. $12x-39y-155=0$

Q.

A line l passing through the origin is perpendicular to the lines
$l_1:(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k},-\infty <t<\infty l_2:(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k},-\infty <s<\infty$
Then, the coordinate(s) of the point(s) on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of l and $l_1$ is (are)

1. $(\frac{7}{3},\frac{7}{3},\frac{5}{3})$

2. $-1,-1,0$

3. $(1,1,1)$

4. $(\frac{7}{9},\frac{7}{9},\frac{8}{9})$

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