# Maharashtra Board 12th Maths Solutions Chapter 6 Line and Plane Ex 6.4

Balbharti 12th Maharashtra State Board Maths Solutions Book Pdf Chapter 6 Line and Plane Ex 6.4 Questions and Answers.

## Maharashtra State Board 12th Maths Solutions Chapter 6 Line and Plane Ex 6.4

Question 1.
Find the angle between planes $$\bar{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})$$ = 13 and $$\bar{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})$$ = 31 .
Solution:
The acute angle θ between the planes $$\bar{r} \cdot \bar{n}_{1}$$ = d1 and $$\bar{r} \cdot \bar{n}_{2}$$ = d2 is given by

Question 2.
Find the acute angle between the line $$\bar{r} \cdot(\hat{i}+2 \hat{j}+2 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}-6 \hat{k})$$ and the plane $$\bar{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})$$ = 0
Solution:
The acute angle θ between the line $$\bar{r}=\bar{a}+\lambda \bar{b}$$ and the plane $$\bar{r} \cdot \bar{n}$$ = d is given by

Question 3.
Show that lines $$\bar{r}=(2 \hat{j}-3 \hat{k})+\lambda(\hat{i}+2 \hat{j}+3 \hat{k})$$ and $$\bar{r}=(2 \hat{i}+6 \hat{j}+3 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+4 \hat{k})$$ are coplanar. Find the equation of the plane determined by them.
Solution:

= 2(-1) + 6(2) + 3(-1)
= -2 + 12 – 3 = 7
∴ $$\bar{a}_{1} \cdot\left(\bar{b}_{1} \times \bar{b}_{2}\right)=\bar{a}_{2} \cdot\left(\bar{b}_{1} \times \bar{b}_{2}\right)$$
Hence, the given lines are coplanar.
The plane determined by these lines is given by
∴ $$\bar{r} \cdot\left(\overline{b_{1}} \times \overline{b_{2}}\right)=\overline{a_{1}} \cdot\left(\overline{b_{1}} \times \overline{b_{2}}\right)$$
i.e. $$\bar{r} \cdot(-\hat{i}+2 \hat{j}-\hat{k})$$
Hence, the given lines are coplanar and the equation of the plane determined by these lines is
$$\bar{r} \cdot(-\hat{i}+2 \hat{j}-\hat{k})$$ = 7

Question 4.
Find the distance of the point $$4 \hat{i}-3 \hat{j}+\hat{k}$$ from the plane $$\bar{r} \cdot(2 \hat{i}+3 \hat{j}-6 \hat{k})$$ = 21 .
Solution:
The distance of the point A($$\bar{a}$$) from the plane $$\bar{r} \cdot \bar{n}=p$$ is given by d = $$\frac{|\bar{a} \cdot \bar{n}-p|}{|n|}$$ …(1)
Here, $$\bar{a}=4 \hat{i}-3 \hat{j}+\hat{k}$$, $$\bar{n}=2 \hat{i}+3 \hat{j}-6 \hat{k}$$, p = 21
∴ $$\bar{a} \cdot \bar{n}$$ = $$(4 \hat{i}-3 \hat{j}+\hat{k}) \cdot(2 \hat{i}+3 \hat{j}-6 \hat{k})$$
= (4)(2) + (-3)(3) + (1)(-6)
= 8 – 9 – 6 = -7
Also, $$\sqrt{2^{2}+3^{2}+(-6)^{2}}=\sqrt{49}$$ = 7
∴ from (1), the required distance
= $$\frac{|-7-21|}{7}$$ = 4units

Question 5.
Find the distance of the point (1, 1, -1) from the plane 3x + 4y – 12z + 20 = 0.
Solution:
The distance of the point (x1, y1, z1) from the plane ax + by + cz + d = 0 is $$\left|\frac{a x_{1}+b y_{1}+c z_{1}+d}{\sqrt{a^{2}+b^{2}+c^{2}}}\right|$$
∴ the distance of the point (1, 1, -1) from the plane 3x + 4y – 12z + 20 = 0 is $$\left|\frac{3(1)+4(1)-12(-1)+20}{\sqrt{3^{2}+4^{2}+(-12)^{2}}}\right|$$
= $$\left|\frac{3+4+12+20}{\sqrt{9+16+144}}\right|=\frac{39}{\sqrt{169}}$$
= $$\frac{39}{13}$$ = 3units