Equation of Line: Symmetrical Form

Q.

On comparing the ratios$\frac{a_1}{a_2},\frac{b_1}{b_2},\frac{c_1}{c_2}$, find out whether the following pairs of linear equations are consistent or inconsistent:

$$\begin{gathered} 3x+2y=5 \\ 2x-3y=7 \end{gathered}$$

Q.

If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then the value of k is -

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

2. $\frac{9}{2}$

3. $-\frac{2}{9}$

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

Q.

The equation of the plane through the point $(2,-1,-3)$ and parallel to the lines $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z}{-4}\text{and}\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}$ is

1. $8x+14y+13z+37=0$
2. $8x-14y-13z+37=0$
3. $8x-14y-13z-37=0$
4. $8x-14y+13z+37=0$
5. $x+2y+2z+6=0$

Q. If the line joining the points $(k,1,2),(3,4,6)$ is parallel to the line joining the points $(-4,3,-6),(5,12,l)$ then $(k,l)$ is:

1. $(-2,7)$
2. $(0,6)$
3. $(0,-6)$
4. $(2,-7)$

Q.

Find the value(s) of $p$ in the following pair of equations: $3x-y-5=0$ and $6x-2y-p=0$, if the lines represented by these equations are parallel.

Q. The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is:

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

Q. Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $xy$- plane has coordinates :

1. $(2,-4,-7)$
2. $(2,4,7)$
3. $(2,-4,-7)$
4. $(-2,4,7)$

Q.

The pair of equations $5x-15y=8$ and $3x-9y=\frac{24}{5}$ has,

1. One solution

2. Two solutions

3. Infinitely many solutions

4. no solution

Q.

The equation of the line which passes through the point (1, 1, 1) and intersecting the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ is

1. $\frac{x-1}{4}=\frac{y-1}{11}=\frac{z-1}{13}$

2. $\frac{x-1}{17}=\frac{y-1}{-3}=\frac{z-1}{17}$

3. $\frac{x-1}{13}=\frac{y-1}{5}=\frac{z-1}{-2}$

4. $\frac{x-1}{3}=\frac{y-1}{10}=\frac{z-1}{17}$

Q. The equation of the plane passing through the point $(-1,2,0)$ and parallel to the lines $L_1:\frac{x}{3}=\frac{y+1}{0}=\frac{z-2}{-1}$ and $L_2:\frac{x-1}{1}=\frac{2y+1}{2}=\frac{z+1}{-1}$ is

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

Q. The coordinate of foot of the perpendicular drawn from the point $A(1,0,3)$ to the line joining the points $B(4,7,1)$ and $C(3,5,3)$ is

1. $(1,2-2)$
2. $(\frac{5}{3},\frac{7}{3},\frac{17}{3})$
3. $(\frac{1}{3},\frac{2}{3},\frac{-2}{3})$
4. $(\frac{-5}{3},\frac{-7}{3},\frac{-17}{3})$

Q. The equation of the line passing through the point (1, 1, –1) and perpendicular to the plane x – 2y – 3z = 7 is

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

Q. The equation of line $AB$ is $\frac{x}{2}=\frac{y}{-3}=\frac{z}{6}$. Through a point $P(1,2,5)$, line $PN$ is drawn perpendicular to $AB$ and line $PQ$ is drawn parallel to the plane $3x+4x+5z=0$ to meet $AB$ at $Q$. Then

1. Coordinates of $N$ are $(\frac{52}{49},-\frac{78}{49},\frac{156}{49})$
2. Equation of line $NQ$ is $3(x-3)=-(2y+9)=z-9$
3. Equation of line $NQ$ is $3(x-3)=(2y+9)=z-9$
4. Coordinates of $Q$ are $(3,\frac{9}{2},9)$

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

1. $\frac{x-1}{2}=\frac{y}{-1}=\frac{z-2}{7}$
2. $\frac{x-1}{1}=\frac{y}{-2}=\frac{z-2}{7}$
3. $\frac{x-1}{2}=\frac{y}{1}=\frac{z-2}{-7}$
4. $\frac{x-1}{1}=\frac{y}{2}=\frac{z-2}{7}$

Q. The point on the line $\frac{x-2}{1}=\frac{y+3}{-2}=\frac{z-5}{-2}=r\left(say\right)$ at a distance of 6 units from the point (2, –3, 5) is

1. (3, –5, –3)
2. (4, –7, –9)
3. (0, 2, –1)
4. (3, 2, 1)

Q. The reflection of the plane $ax+by+cz+d=0$ in the plane $a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime }=0$ is:

1. $(a^{\prime 2}+b^{\prime 2}+c^{\prime 2})(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })=(a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime })(ax+by+cz+d)$
2. $2(a^{\prime 2}+b^{\prime 2}+c^{\prime 2})(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })=(a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime })(ax+by+cz+d)$
3. $(a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime })(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })=(a^{\prime 2}+b^{\prime 2}+c^{\prime 2})(ax+by+cz+d)$
4. $2(a{a}^{\prime }+b{b}^{\prime }+c{c}^{\prime })(a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime })=(a^{\prime 2}+b^{\prime 2}+c^{\prime 2})(ax+by+cz+d)$

Q. A line whose direction cosines is proportional to $2,1,2$, meets with the lines $x=y+a=z$ and $x+a=2y=2z$ at $A$ and $B$ respectively. If the distance between $A$ and $B$ is $d$, then $\frac{12d}{|a|}$ is

Q. The Cartesian equation of a line is . Write its vector form.

Q.

Direction ratios of the line represented by the equation x = ay + b, z = cy + d are

1. (a, 1, c)

2. (a, b - d, c)

3. (c, 1, a)

4. (b, ac, d)

Q.

If the equation of a line and a plane be $\frac{x+3}{2}=\frac{y-4}{3}=\frac{z+5}{2}$ and 4x-2y-z=1 respectively,

1. Line is parallel to the plane

2. Line is perpendicular to the plane

3. Line lies in the plane

4. None of these

Q.

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}and\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$ perpendicular, then find the value of k.

Q.

The equation of the line which passes through the point (1, 1, 1) and intersecting the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x+2}{1}=\frac{y-3}{2}=\frac{z+1}{4}$ is

1. $\frac{x-1}{4}=\frac{y-1}{11}=\frac{z-1}{13}$

2. $\frac{x-1}{17}=\frac{y-1}{-3}=\frac{z-1}{17}$

3. $\frac{x-1}{13}=\frac{y-1}{5}=\frac{z-1}{-2}$

4. $\frac{x-1}{3}=\frac{y-1}{10}=\frac{z-1}{17}$

Q. Let image of the line $\frac{x-1}{3}=\frac{y-3}{5}=\frac{z-4}{2}$ in the plane $2x-y+z+3=0$ be $L.$ If a plane $7x+py+qz+r=0$ $(p,q,r\in R)$ is such that it contains the line $L$ and perpendicular to the plane $2x-y+z+3=0,$ then the value of $(p+3q+r)$ is

Q.

Find the equation of line through $(-4,1,3)$ and parallel to the plane $x+y+z=3$ while the line intersects another line whose equation is $x+y+z=0=x+2y-3z+5$.

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

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

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

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

Q. If the image of the point $P(1,-2,3)$ in the plane, $2x+3y-4z+22=0$ measured parallel to the line, $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ is $Q,$ then $PQ$ is equal to

1. $3\sqrt{5}$
2. $6\sqrt{5}$
3. $2\sqrt{42}$
4. $\sqrt{42}$

Q. If $\frac{x-1}{l}=\frac{y-2}{m}=\frac{z+1}{n}$ is the equation of the line through (1, 2, -1) and (-1, 0, 1), then (l, m, n) is
[MP PET 1992]

1. (1, 2, -1)

2. (-1, 0, 1)

3. (0, 1, 0)

4. (1, 1, -1)

Q.

Find the distance between the planes $r(2\hat{i}-\hat{j}+3\hat{k})-4=0$ and $r(6\hat{i}-3\hat{j}+9\hat{k})+13=0$.

1. $\frac{5}{3\sqrt{14}}$

2. $\frac{10}{3\sqrt{14}}$

3. $\frac{25}{3\sqrt{14}}$

4. None of these

Q.

Find the value of p so that the lines $\frac{1-x}{3}=\frac{7y-14}{2p}=\frac{z-3}{2}and\frac{7-7x}{3p}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles

Q. Equation of a line passing through (2, -1, 1) and parallel to the line whose equation is $\frac{x-3}{2}=\frac{y+1}{7}=\frac{z-2}{-3}$is

1. $\frac{x-2}{3}=\frac{y+1}{-1}=\frac{z-1}{2}$
2. $\frac{x-2}{2}=\frac{y+1}{7}=\frac{z-1}{-3}$
3. $\frac{x-2}{2}=\frac{y-1}{-1}=\frac{z+3}{1}$
4. $\frac{x-3}{2}=\frac{y+1}{-1}=\frac{z-2}{1}$

Q. If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, then k =

1. $\frac{2}{9}$
2. $\frac{9}{2}$
3. 0
4. 3