KC Sinha Solution Class 12 Chapter 30 3D Geometry : Plane (त्रिविमीय ज्यामिति : समतल ) Exercise 30.1 (Q41-Q50)

 Exercise 30.1

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 :