Perpendicular Distance of a Point from a Plane

Q.

The equation of the line parallel to the line $2x-3y=1$ and passing through the middle point of the line segment joining the points $(1,3)$ and $(1,-7)$, is:

1. $2x-3y+8=0$

2. $2x-3y=8$

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

4. $2x-3y=4$

Q. If $P_1$ and $P_2$ are the lengths of the perpendiculars from the points (2, 3, 4) and (1, 1, 4) respectively from the plane 3x-6y+2z+11 =0, then $P_1$ and $P_2$ are the roots of the equation

1. $P^2-23P+7=0$
2. $7P^2-23P+16=0$
3. $P^2-17P+16=0$
4. $P^2-16P+7=0$

Q. Let $F$ be the foot of perpendicular from $P(1,2,-3)$ on the line $L:\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}.$ Given $Q(x_1,y_1,z_1)$ and $R(x_2,y_2,z_2)$ are points at a distance of $3$ units from $F$ on line $L.$ Let ${\pi }_1$ and ${\pi }_2$ be two planes passing through $Q$ and $R$ respectively and perpendicular to the line of intersection of the planes $9x-7y+6z+48=0$ and $x+y-z=7.$ Then which of the following statements is (are) CORRECT?

1. The coordinates of $F$ are $(1,1,-1)$
2. The value of $\sum _{i=1}^2(x_i^2+y_i^2+z_i^2)$ equals $24$
   
3. The distance between planes ${\pi }_1$ and ${\pi }_2$ equals $\frac{88}{\sqrt{482}}$
4. The normal vector of planes ${\pi }_1$ and ${\pi }_2$ is parallel to the vector $\hat{i}+15\hat{j}+16\hat{k}.$

Q.

A plane passes through (1, -2, 1) and is perpendicular to two planes $2x-2y+z=0$ and $x-y+2z=4,$ then the distance of the plane form the point (1, 2, 2) is

1. 0

2. 1

3. $\sqrt{2}$

4. $2\sqrt{2}$

Q.

A plane passes through (1, -2, 1) and is perpendicular to two planes $2x-2y+z=0$ and $x-y+2z=4,$ then the distance of the plane form the point (1, 2, 2) is

1. 0

2. 1

3. $\sqrt{2}$

4. $2\sqrt{2}$

Q. Given $\overrightarrow{\alpha }=3\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{\beta }=\hat{i}-2\hat{j}-4\hat{k}$ are the position vectors of the points $A$ and $B.$ Then the distance of the point $-\hat{i}+\hat{j}+\hat{k}$ from the plane passing through $B$ and perpendicular to $AB$ is

1. $5$
2. $10$
3. $15$
4. $20$

Q. A rod of length $2$ units whose one end is $(1,0,-1)$ and other end touches the plane $x-2y+2z+4=0$. Then

1. the rod sweeps a figure whose volume is $\pi$ cubic units.
2. the area of the figure which the rod traces on the plane is $2\pi$ units.
3. the length of projection of the rod on the plane is $\sqrt{3}$ units.
4. the centre of the region which the rod traces on the plane is $(\frac{2}{3},\frac{2}{3},\frac{-5}{3})$

Q. Given $\overrightarrow{\alpha }=3\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{\beta }=\hat{i}-2\hat{j}-4\hat{k}$ are the position vectors of the points $A$ and $B.$ Then the distance of the point $-\hat{i}+\hat{j}+\hat{k}$ from the plane passing through $B$ and perpendicular to $AB$ is

1. $5$
2. $10$
3. $15$
4. $20$