KC Sinha: Exercise 30.1- Mathematics Solution Class 12 Chapter 30 त्रिविमीय ज्यामिति : समतल

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Question 1

Find the cartesian equations of the following planes whose vector equations are

(i) $\vec{r} \cdot(3 \hat{i}+3 \hat{j}-4 \hat{k})=0$

(ii) $\vec{r} \cdot(2 \hat{i}-7 \hat{j}+4 \hat{k})+1=0$

(iii) $\vec{r} \cdot(\hat{i}+\hat{j}-\hat{k})=2$

(iv) $\vec{r} \cdot[(s-2 t) \hat{i}+(3-t) \hat{j}+(2 s+t) \hat{k}]=15$

Sol :

Question 2

Find the vector equation of the following planes whose cartesian equations are

(i) $2x+3 y-z-1=0$

(ii) $x+2 y+3 z+5=0$

(iii) $x-3 y+6 z=0$

Sol :

Question 3

Find the equation of the plane with intercepts 2 , 3 and 4 on the x, y and z-axes respectively.

Sol :

Question 4

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

Sol :

Question 5

Find the equation of the plane which cuts intercepts 2, 3 -4  on the axes.

Sol :

Question 6

Find the intercepts of the plane 3x+4y-7z=84 on the axes. Also find the length of perpendicular from origin to this line and direction cosines of this normal.

Sol :

Question 7

Find the intercepts cut off on the axes by the plane 2x+y-z=5.

Sol :

Question 8

Find the equation of the plane which meets the axes in A, B, C given that the centroid of ΔABC is the point (α ,β ,ɣ).

Sol :

Question 9

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector $3 \hat{i}+5 \hat{j}-6 \hat{k}$.

Sol :

Question 10

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

Sol :

Question 11

In each of the following cases, determine the direction cosines of the normal to the plane and its distance from the origin

(i) $2 x-3 y+4 z-6=0$

(ii) $2 x+3 y-z=5$

(iii) $x+y+z=1$

(iv) $5 y+8=0$

(v) $z=2$

Sol :

Question 12

Find the direction cosines of the unit vector perpendicular to the plane $\vec{r} \cdot(6 \hat{i}-3 \hat{j}-2 \hat{k})+1=0$ passing through the origin.

Sol :

TYPE-III

Question 13

Find the angle between the planes whose vector equations are $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=5$ and $\vec{r} \cdot(3 \hat{i}-3 \hat{j}+5 \hat{k})=3$

Sol :

Question 14

Find the angle between the planes

(i) $2 x-y+z=6$ and $x+y+2 z=7$.

(ii) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$

(iii) $3 x-6 y+2 z=7$ and $2 x+2 y-2 z=5$

(iv) $2 x+y-2 z=5$ and $3 x-6 y-2 z=7$

Sol :

Question 15

Determine whether the following pair of planes are parallel or perpendicular, and in case they are neither, find the angle between them.

(i) $2 x-y+3 z-1=0$ and $2 x-y+3 z+3=0$

(ii) $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$

(iii) $2 x+y+3 z-2=0$ and $x-2 y+5=0$

(iv) $4 x+8 y+z-8=0$ and $y+z-4=0$

(v) $3 x-4 y+5 z=0$ and $2 x-y-2 z=5$

Sol :

Question 16

Find the angle between the line $\frac{x+1}{2}=\frac{y}{3}=\frac{z-3}{6}$ and the plane $10 x+2 y-11 z=3$

Sol :

TYPE-IV

Question 17

(i) Find the equation of the plane containing point (1,-1,2) and perpendicular to each of the planes 2x+3y-2z=5 and x+2y-3z=8

Sol :

(ii) Find the equation of the plane passing through the point (-1,-1,2) and perpendicular to each of the planes : 2x+3y-3z=2 and 5x-4y+z=6

Sol :

Question 18

Find the equation of the plane passing through the point (-1,3,2) and perpendicular to each of the planes x+2y+3z=5 and 3x+3y+z=0.

Sol :

Question 19

Find the vector and cartesian equations of the planes

(i) that passes through the point (1, 4, 6) and the normal vector to the plane is $\hat{i}-2 \hat{j}+\hat{k}$

(ii) that passes through the point (1,0,-2) and normal vector to the plane is $\hat{i}+\hat{j}-\hat{k}$

Sol :

Question 20

If O be the origin and the coordinates of P be (1,2,-3) then find the equation of the plane passing through P and perpendicular to OP

Sol :

Question 21

Find the equation of the plane through (3,4,-1) which is parallel to the plane $\vec{r} \cdot(2 \hat{i}-3 \hat{j}+5 \hat{k})+7=0$

Sol :

Question 22

Find the equation of the plane passing through (a, b, c) and parallel to the plane $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=2$.

Sol :

Question 23

(i) Find the equation of the plane passing through the point (3,-3,1) and perpendicular to the line joining (3,4,-1) and (2,-1,5).

(ii) Find the equation of the plane passing through the points (3,4,1) and (0,1,0) and parallel to the line $\frac{x+3}{2}=\frac{y-3}{7}=\frac{z-2}{5}$

Sol :

Question 24

Find the vector and cartesian equation of the plane which passes through the point (5,2,-4) and perpendicular to the line with direction ratios (2,3,-1).

Sol :

Question 25

Find the equation of the plane through the point (1,4,-2) and parallel to the plane -2x+y-3z=7

Sol :

Question 26

Find the equation of the plane through the points (2,-3,1) and (5,2,-1) and perpendicular to the plane x-4y+5z+2=0.

Sol :

Question 27

Find the equation of the plane through the points (-1,1,1) and (1,-1,1) and perpendicular to the plane x+2y+2z=5.

Sol :

Question 28

Find the equation of the plane passing through the point (1,1,-1) and perpendicular to the planes x+2y+3z-7=0 and 2x-3y+4z=0.

Sol :

TYPE-V

Question 29

Find the equation of a plane through the points (2,1,0), (3,-2,-2) and (3,1,7).

Sol :

Question 30

Find the equation of the plane that passes through three points (1,1,0) , (1,2,1), (-2,2,-1).

Sol :

Question 31

Find the vector equation of the plane passing through the points (2,5,-3), (-2,-3,5),(5,3,-3)

Sol :

Question 32

Can there be a unique equation of the plane passing through points (2,5,-3), (-2,-3,5) and (5,3,-3). Give reasons for your answer.

Sol :

TYPE-VI

Question 33

Find the coordinates of the point where the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z+3}{4}$ meets the plane x+y+4z=6.

Sol :

Question 34

(i) Find the coordinates of the point where the line through $(3,-4,-5)$ and $(2,-3,1)$ crosses the plane 2x+y+z=7

(ii) Find the coordinates of the point where the line through the points A(3,4,1) and B(5,1,6) crosses the xy-plane.

(iii) Find the coordinates of the point where the line through (5,1,6) and (3,4,1) crosses the zx-plane.

Sol :

Question 35

Find the coordinates of the point where the line through (5,1,6) and (3,4,1) crosses the yz-plane.

Sol :

Question 36

Find the distance of the point (-1,-5,-10) from the point of intersection of the line $\vec{r}=2 \hat{i}-\hat{j}+2 \hat{k}+\lambda(3 \hat{i}+4 \hat{j}+2 \hat{k})$ and the plane $\vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=5$

Sol :

Question 37

Find the coordinates of the foot of perpendicular drawn from origin to the planes

(i) x+y+z=1

(ii) 3y+4z-6=0

(iv) 2x+3y+4z-12=0

(v) 2x-3y+4z-6=0

(iii) 5y+8=0

Sol :

Question 38

(i) Find the image of the point (2,-3,4) with respect to the plane 4x+2y-4z+3=0.

(ii) Find the image of the point (1,3,4) in the plane 2x-y+z+3=0.

(iii) From the point P(1,2,4) a perpendicular in drawn on the plane 2x+y-2z+3=0. Find the equation, the length and the coordinates of the foot of perpendicular.

Sol :

TYPE-VII

Question 39

Find the equation of the plane passing through the intersection of the planes $\vec{r} \cdot(2 \hat{i}+\hat{j}+3 \hat{k})=7, \vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})=9$ and the point (2,1,3)

Sol :

Question 40

Find the equation of the plane passing through the intersection of the planes $\vec{r} \cdot(2 \hat{i}+\hat{j}+3 \hat{k})=7, \vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})=9$ and the point (3,2,-1).

Sol :

Question 41

Find the vector equation of the plane through the line of intersection of the planes $\vec{r} \cdot(2 \hat{i}+2 \hat{j}-3 \hat{k})=7, \vec{r} \cdot(2 \hat{i}+5 \hat{j}+3 \hat{k})=9$ and through the point (2,1,3)

Sol :

Question 42

Find the vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=6$ and $\vec{r} \cdot(2 \hat{i}+3 \hat{j}+4 \hat{k})=-5$ and the point (1,1,1).

Sol :

Question 43

Find the equation of the plane through the intersection of the planes 3x-y+2z-4=0 and x+y+z-2=0 and the point (2,2,1).

Sol :

Question 44

Find the equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$ which is perpendicular to the plane $x-y+z=0$

Question 45

Find the equation of the plane passing through the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r} \cdot(2 \hat{i}+3 \hat{j}-\hat{k})+4=0$ and parallel to x-axis.

Sol :

Question 46

Find the equation of the plane which contains the line of intersection of the planes $\vec{r} \cdot(\hat{i}+2 \hat{j}+3 \hat{k})-4=0, \vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})+5=0$ and which is perpendicular to the plane $\vec{r} \cdot(5 \hat{i}+3 \hat{j}-6 \hat{k})+8=0$.

Sol :

Question 47

Show that the lines $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$ and $\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$ are coplanar.

Sol :

Question 48

Show that the lines $\frac{x-3}{2}=\frac{y+1}{-3}=\frac{z+2}{1}$ and $\frac{x-7}{-3}=\frac{y}{1}=\frac{z+7}{2}$ are coplanar. Also find the equation of the plane containing them.

Sol :

Question 49

Show that the lines $\frac{x-a+d}{\alpha-\delta}=\frac{y-a}{\alpha}=\frac{z-a-d}{\alpha+\delta}$ and $\frac{x-b+c}{\beta-\gamma}=\frac{y-b}{\beta}=\frac{z-b-c}{\beta+\gamma}$ are coplanar.

Sol :

Type-IX

Question 50

Find the distance of each of the following points from the corresponding given planes

(i) (-6,0,0) ; 2x-3y+6z-2=0

(ii) (2,3,-5) ; x+2y-2z=9

(iii) (0,0,0) ; 3x-4y+12z=3

(iv) (3,-2,1); 2x-y+2z+3=0

Sol :

Question 51

Find the distance of a point (2,5,-3) from the plane $\vec{r} \cdot(6 \hat{i}-3 \hat{j}+2 \hat{k})=4$

Sol :

Question 52

If a plane has intercepts a, b, c on axes and is at a distance of p units from the origin, then prove that

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

Sol :

Question 53

Find the distance between the point P(6,5,9) and the plane determined by the points A(3,-1,2), B(5,2,4) and C(-1,-1,6)

Sol :

Question 54

Find the distance between the planes 2x+3y+4z=4 and 4x+6y+8z=12.

Sol :

Question 55

Find the equation of the line of intersection of the planes x-2y+z=1 and x+2y-2z=5 in symmetric form.

Sol :

TYPE-X

Question 56

Find the equation of the line through point (1,2,3) and parallel to line x-y+2z=5, 3x+y+z=6

Sol :

Question 57

Prove that the lines x=ay+b, z=cy+d and $x=a^{\prime} y+b, z=c^{\prime} y+d^{\prime}$ are mutually perpendicular if $a a^{\prime}+c c^{\prime}=-1$

Sol :

Question 58

Find the vector equation of the line passing through (1,2,3) and parallel to the planes $\vec{r} \cdot(\hat{i}-\hat{j}+2 \hat{k})=5$ and $\vec{r} \cdot(3 \hat{i}+\hat{j}+\hat{k})=6$.

Sol :