Class 12 RD Sharma Solutions- Chapter 29 The Plane - Exercise 29.15 | Set 2

Question 8. Find the image of the point (1, 3, 4) in the plane 2x - y + z + 3 = 0.

Solution:

According to the question we have to find the image of point P(1, 3, 4)

in the plane 2x - y + z +3 = 0

Now let us assume that Q be the image of the point.

Here, the direction ratios of normal to plane = 2, -1, 1

The direction ratios of PQ which is parallel to the normal to the plane 

is proportional to 2, -1, 1 and the line PQ is passing through point P(1, 3, 4).

Thus, equation of the line PQ is: 

\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\\ \frac{x-1}{2}=\frac{y-3}{-1}=\frac{z-4}{-1}=λ

Now, the general point on the line PQ = (2λ + 1, -λ + 3, λ + 4)

Let Q = (2λ + 1, -λ + 3, λ + 4)          -Equation(1)

Here, Q is the image of P, so R is the mid point of PQ

Coordinates of R = \left(\frac{2λ+1+1}{2},\frac{-λ+3+3}{2},\frac{λ+4+4}{2}\right)\\ =\left(\frac{2λ+2}{2},\frac{-λ+6}{2},\frac{λ+8}{2}\right)\\ =\left(λ+1,\frac{-λ+6}{2},\frac{λ+8}{2}\right)

Point R is lies on the plane 2x - y + z + 3 = 0

= 2(λ + 1) - \left(\frac{-λ+6}{2}+\frac{λ+8}{2}\right) = 0

4λ + 4 + λ - 6 + λ + 8 + 6 = 0

6λ = -12

λ = -2

Now, put the value of λ in equation(1), we get

= (-4 + 1, 2 + 3, -2 + 4)

= (-3, 5, 2)

Hence, the image of point P(1, 3, 4) is (-3, 5, 2)

Question 9. Find the distance of the point with position vector -\hat{i}-5\hat{j}-10\hat{k}    from the point of intersection of the line \vec{r}=(2\hat{i}-\hat{j}+2\hat{k})+λ(3\hat{i}+4\hat{j}+12\hat{k})    with the plane \vec{r}.(\hat{i}-\hat{j}+\hat{k})=5  .

Solution:

According to the question we have to find distance of a point A with position 

vector (-\hat{i}-5\hat{j}-10\hat{k})    from the point of intersection of 

line \vec{r}=(2\hat{i}-\hat{j}+2\hat{k})+λ(3\hat{i}+4\hat{j}+12\hat{k})  

with plane \vec{r}.(\hat{i}-\hat{j}+\hat{k})=5  

Let the point of intersection of line and plan be B(\vec{b})

The line and the plane will intersect when,

[(2\hat{i}-\hat{j}+2\hat{k})+λ(3\hat{i}+4\hat{j}+12\hat{k})](\hat{i}-\hat{j}+\hat{k})=5\\ [(2+3λ)\hat{i}+(-1+4λ)\hat{j}+(2+12λ)\hat{k}](\hat{i}-\hat{j}+\hat{k})=5\\

(2 + 3λ)(1) + (-1 + 4λ)(-1) + (2 + 12λ)(1) = 5

2 + 3λ + 1 - 4λ + 2 + 12λ = 5

11λ = 5 - 5

λ = 0

So, the point B is given by

\vec{b}=(2\hat{i}-\hat{j}+2\hat{k})+(0)(3\hat{i}+4\hat{j}+12\hat{k})\\ \vec{b}=(2\hat{i}-\hat{j}+2\hat{k})

\vec{AB}=\vec{b}-\vec{a}

=(2\hat{i}-\hat{j}+2\hat{k})-(-\hat{i}-5\hat{j}-10\hat{k})

=(2\hat{i}-\hat{j}+2\hat{k}+\hat{i}+5\hat{j}+10\hat{k})=(3\hat{i}+4\hat{j}+12\hat{k})\\ |\vec{AB}|=\sqrt{(3)^2+(4)^2+(12)^2}=\sqrt{9+16+144}=\sqrt{169}=13

The required distance is 13 units.

Question 10. Find the length and the foot of the perpendicular from the point (1, 1, 2) to the plane \vec{r}.(\hat{i}-2\hat{j}+4\hat{k})+5=0  .

Solution: 

Plane = x - 2y + 4z + 5 = 0          -Equation(1)

Point = (1, 1, 2)

D = \left|\frac{1-2+8+5}{\sqrt{1+4+16}}\right|

= 12/√21

The length of the perpendicular from the given point to the plane = 12/√21

Let us assume that the foot of perpendicular be (x, y, z). 

So DR's are in proportional

\frac{x-1}{1}=\frac{y-1}{-2}=\frac{z-2}{4}=k

x = k + 1

y = -2k + 1

z = 4k + 2

Substitute (x, y, z) = (k + 1, -2k + 1, 4k + 2) in the plane equation(1)

k + 1 + 4k - 2 + 16k + 8 + 5 = 0

21k = -12

k = -12/21 = -4/7 

Hence, the coordinate of the foot of the perpendicular  = (3/7, 15/7, -2/7) 

Question 11. Find the coordinates of the foot of the perpendicular and the perpendicular distance of the point P(3, 2, 1) from the plane 2x - y + z + 1 = 0. Find also the image of the point in the plane.

Solution: 

Given:

Plane = 2x - y + z + 1 = 0          -Equation(1)

Point P = (3, 2, 1)

D =\left|\frac{6-2+1+1}{\sqrt{4+1+1}}\right|=\frac{6}{\sqrt{6}}=\sqrt{6}

The perpendicular distance of the point P from the plane(D) = √6

Let us assume that the foot of perpendicular be (x, y, z). 

So DR's are in proportional

\frac{x-3}{2}=\frac{y-2}{-1}=\frac{z-1}{1}=k

x = 2k + 3

y = -k + 2

z = k + 1

Substitute (x, y, z) = (2k + 3, -k + 2, k + 1) in the plane equation(1)

4k + 6 + k - 2 + k + 1 + 1 = 0

6k = -6

k = -6/6 = -1

The coordinate of the foot of the perpendicular = (1, 3, 0) 

Question 12. Find the direction cosines of the unit vector perpendicular to the plane \vec{r}.(6\hat{i}-3\hat{j}-2\hat{k})+1=0     passing through the origin.

Solution: 

Given:

Equation of the plane \vec{r}.(6\hat{i}-3\hat{j}-2\hat{k})+1=0

Thus, the direction ratios normal to the plane are 6, -3 and -2

Hence, the direction cosines to the normal to the plane are

= \frac{6}{\sqrt{6^2+(-3)^2+(-2)^2}},\frac{-3}{\sqrt{6^2+(-3)^2+(-2)^2}},\frac{-2}{\sqrt{6^2+(-3)^2+(-2)^2}}\\ 

= 6/7, -3/7, -2/7

= -6/7, 3/7, 2/7

The direction cosines of the unit vector perpendicular to the plane 

are same as the direction cosines of the unit vector perpendicular

to the plane are: -6/7, 3/7, 2/7 

Question 13. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x - 3y + 4z - 6 = 0.

Solution: 

According to the question,

Plane = 2x - 3y + 4z - 6 = 0

The direction ratios of the normal to the plane are 2, -3 and 4.

Thus, the direction ratios of the line perpendicular to the plane are 2, -3 and 4.

The equation of the line passing (x₁, y₁, z₁) having direction ratios a, b and c is

 \frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}

Thus, the equation of the line passing through the origin 

with direction ratios 2, -3 and 4 is

\frac{x-0}{2}=\frac{y-0}{-3}=\frac{z-0}{4}\\ \frac{x}{2}=\frac{y}{-3}=\frac{z}{4}=r      

Here, r is same constant.

Any point on the line is of the form 2r, -3r, and 4r, 

if the point P(2r, -3r, 4r) lies on the plane 2x - 3y + 4z - 6 = 0.

Thus, we have,

2(2r) - 3(-3r) + 4(4r) - 6 = 0

4r + 9r + 16r - 6 = 0

29r = 6

r = 6/29

Thus, the coordinates of the point of intersection of the perpendicular 

from the origin and the plane are:

P(2×6/29, -3×629, 4×6/29) = P(12/29, -18/29, 24/29) 

Question 14. Find the length and the foot of the perpendicular from the point (1, 3/2, 2) to the plane 2x - 2y + 4z +5 = 0.

Solution:

Given: 

Point = (1, 3/2, 2) 

Plane = 2x - 2y + 4z + 5 = 0

D = \left|\frac{2-3+8+5}{\sqrt{4+4+16}}\right|=\frac{12}{2\sqrt{6}}

= √6

So, the length of the perpendicular from the point to the plane(D) = √6

Let the foot of perpendicular be (x, y, z). So, DR's are in proportional 

\frac{x-1}{2}=\frac{y-\frac{3}{2}}{-2}=\frac{z-2}{4}=k

x = 2k + 1

y = -2k + 3/2

z = 4k + 2

So, using the values of x, y, z in equation of the plane we have,

2(2k + 1) - 2(-2k + 2/3) +4(4k + 2) + 5 = 0

4k + 2 + 4k - 3 + 16k + 8 + 5 = 0

24k = -12

k = -1/2

So, the coordinate of the foot of the perpendicular = (0, 5/2, 0)

Summary

Exercise 29.15 in RD Sharma Class 12 typically deals with finding the equation of a plane under various conditions. This may include:

• Finding planes parallel or perpendicular to given planes
• Determining planes passing through given points
• Finding planes containing given lines
• Calculating the distance between parallel planes
• Solving problems involving the angle between planes