Equation of a Plane : Normal Form

Q.

The foot of the perpendicular drawn from the point$(4,2,3)$ to the line joining the points $(1,-2,3)$ and $(1,1,0)$ lies on the plane:

1. $x-y-2z=1$

2. $x-2y+z=1$

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

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

Q.

The distance of origin from the point of intersection of the line $\left(\frac{\mathrm{x}}{2}\right)=\frac{\left[\mathrm{y}-2\right]}{3}=\frac{\left[\mathrm{z}-3\right]}{4}$ and the plane $2\mathrm{x}+\mathrm{y}-\mathrm{z}=2$ is

1. $\sqrt{120}$

2. $\sqrt{83}$

3. $2\sqrt{19}$

4. $\sqrt{78}$

Q.

A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $(1,-2,2)$ and $(2,3,-1)$ respectively. If this plane also passes through the point $(\alpha ,-3,5)$, then $\alpha$ is equal to

1. $-5$

2. $10$

3. $5$

4. $-10$

Q.

Which of the following is the general equation of plane ?

1. 

2. 

3. 

4. 

Q.

What are the coordinates of the foot of perpendicular from the point $(–1,3)$ to the line $3x-4y-16=0$?

Q.

Equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at the greatest distance from the point (0, 0, 0) is

1. $4x+3y+5z=25$

2. $4x+3y+5z=50$

3. $3x+4y+5z=59$

4. $x+7y-5z=2$

Q. The distance of the plane 2x - 3y + 4z = 6 from the origin is equal to

1. $\frac{2}{\sqrt{29}}$
2. $\frac{3}{\sqrt{29}}$
3. $\frac{5}{\sqrt{29}}$
4. $\frac{6}{\sqrt{29}}$

Q. The equation of a line passing through the point $(4,-2,5)$ and parallel to the vector $3\hat{i}-\hat{j}+2\hat{k}$ in Cartesian form is

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

Q.

If $x$ coordinates of a point $P$ of the line joining the points $Q(2,2,1)\mathrm{and}R(5,2,-2)$ is $4$, then the $z$-co-ordinates of $P$ is:

1. $-2$

2. $-1$

3. $1$

4. $2$

Q.

The Cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$, write its vector form.

Q.

Prove that if a plane has the intercepts a, b, c and if it is at a distance of p units from the origin, then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{p^2}$

Q.

Find the circumcentre of the triangle formed by the points $(2,3)$, $(1,-5)$ and $(-1,4)$

Q.

Find the vector equation of the straight line passing through (1, 2, 3) and perpendicular to the plane $r.(\hat{i}+2\hat{j}-5\hat{k})+9=0$

Q. The perpendicular distance of the origin form the plane which makes intercepts 12, 3 and 4 on x, y, z axes respectively, is

1. 13
2. 11
3. 17
4. $\frac{6\sqrt{2}}{\sqrt{1}3}$

Q. The position vector of a point in which a line through the origin and perpendicular to the plane $2x-y-z=4,$ meets the plane $\overrightarrow{r}\cdot (3\hat{i}-5\hat{j}+2\hat{k})=6$ is

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

Q. The perpendicular distance of the origin form the plane which makes intercepts 12, 3 and 4 on x, y, z axes respectively, is

1. $\frac{6\sqrt{2}}{\sqrt{1}3}$
2. 13
3. 17
4. 11

Q.

Determine the direction cosines of the normal to plane and the distance from the origin:

2x +3y -z = 5

Q. Maximise $Z=3x+4y$
Subject to the constraints :
$x+y\le 4,x\ge 0,y\ge 0.$

Q.

What is the distance of the plane $2x-3y+4z=6$ from the origin?

1. $\frac{2}{\sqrt{29}}$

2. $\frac{3}{\sqrt{29}}$

3. $\frac{5}{\sqrt{29}}$

4. $\frac{6}{\sqrt{29}}$

Q. Equation of the line $L=0$ is given by $2x-y+z-2=0=x+2y-z-3$, then which of the following is/are true?

1. Equation of the plane containing the line $L=0$ and passing through the point $(3,2,1)$ is $x-3y+2z+1=0$
2. Equation of the plane containing the line $L=0$ and passing through the point $(3,2,1)$ is $2x-3y+z-1=0$
3. Vector equation of the plane which passes through the point $(-1,3,2)$ and is perpendicular to the line $L=0$ is $\overrightarrow{r}.(\hat{i}-3\hat{j}-5\hat{k})+20=0$
4. Vector equation of the plane which passes through the point $(-1,3,2)$ and is perpendicular to the line $L=0$ is $\overrightarrow{r}.(\hat{i}+3\hat{j}+5\hat{k})-18=0$

Q. Find the equation of the plane which is at a distance of $\frac{6}{\sqrt{29}}$ from the origin and its normal vector from the origin is $2\hat{i}-3\hat{j}+4\hat{k}$

1. $3x-4y+2z=6$
2. $2x-3y+4z=6$
3. $2x+3y+4z=6$
4. $2x-3y-4z=6$

Q. The minimum distance between the origin and the plane which is perpendicular bisector of the line joining the points $(1,3,5)$ and $(3,7,-1)$, is

1. $\frac{6}{\sqrt{14}}$
2. $\frac{3}{7}$
3. $\frac{3}{\sqrt{14}}$
4. $\frac{6}{7}$

Q. The direction ratios of a normal to the plane passing through $(0,1,1),(1,1,2)$ and $(-1,2,-2)$ are

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

Q. Two points $A(7,0,6)$ and $B(3,4,2)$ lie on a plane $P_1$ and plane $P_2:2x-5y=15$ is perpendicular to the plane $P_1$ and point $(2,\alpha ,\beta )$ also lie on plane $P_1$. Then find the value of $(2\alpha -3\beta )$

1. $12$
2. $17$
3. $5$
4. $7$

Q. If the vectors $x\hat{i}+\hat{j}+\hat{k},\hat{i}+y\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+z\hat{k}$ are co-planar where $x\ne 1,y\ne 1,z\ne 1$, then prove that $\frac{1}{1-x}+\frac{1}{1-y}+\frac{1}{1-z}=1$.

Q. The vector equation of the line passing through the point $(-1,-1,2)$ and parallel to the line $2x-2=3y+1=6z-2,$ is

Q. The perpendicular distance of the origin form the plane which makes intercepts 12, 3 and 4 on x, y, z axes respectively, is

1. 13
2. 11
3. 17
4. $\frac{6\sqrt{2}}{\sqrt{1}3}$

Q.

Find the equation of plane which is at a distance of 8 units from the origin and which is normal to the line having direction ratios 2, 1, 2 ?

1. x + 2y + z = 24

2. 2x + y + 2z = 24

3. 2x + 2y + z = 72

4. x + 2y + 2z = 72

Q. Lines $OA,OB$ are drawn from $O$ with direction cosines proportional to $(1,-2,-1),(3,-2,3).$ Find the direction cosines of the normal to the plane $AOB$

1. $⟨\pm \frac{4}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$
2. $⟨\pm \frac{2}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$
3. $⟨\pm \frac{8}{\sqrt{29}}\pm \frac{6}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$
4. $⟨\pm \frac{8}{\sqrt{29}}\pm \frac{3}{\sqrt{29}}\pm \frac{-2}{\sqrt{29}}⟩$

Q. The perpendicular distance of the origin from the plane which makes intercepts 12, 3 and 4 on x, y, z axes
respectively, is

1. 13
2. 11
3. 17
4. $\frac{6\sqrt{2}}{\sqrt{13}}$