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# Class 12 RD Sharma Solutions – Chapter 29 The Plane – Exercise 29.12

• Last Updated : 28 Mar, 2021

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

Solution:

We know, the equation of line through the points (x1,y1,z1) and (x2,y2,z2) is

Therefore, equation of line joining (5, 1, 6) and (3, 4, 1) is

â‡’

Let, , where  is a constant.

â‡’

Coordinates of any point on the line is in the form of

Since, the line crosses the yz-plane, the point  must satisfy the equation of plane x=0,

â‡’ â‡’

Therefore, coordinates of points is given by, putting  we get,

â‡’

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

Solution:

We know, the equation of line through the points (x1,y1,z1) and (x2,y2,z2) is

Therefore, equation of line joining (5, 1, 6) and (3, 4, 1) is

â‡’

Let,  ,where  is a constant.

â‡’

Coordinates of any point on the line is in the form of

Since, the line crosses the zx-plane, the point  must satisfy the equation of plane y=0,

â‡’  â‡’

Therefore, the coordinates of point is given by, putting  we get,

â‡’

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

Solution:

We know, the equation of line through the points (x1,y1,z1) and (x2,y2,z2) is

Therefore, line joining the points (3, -4, -5) and (2, -3, 1) is

â‡’

Let  where  is constant.

â‡’

The coordinates of any point on the line is given by

The line crosses the plane, therefore, point must satisfy the plane equation.

â‡’

Therefore, The coordinates of point are given by, putting ,

â‡’

â‡’ (1, -2, 7)

### and the plane

Solution:

Given equation of line is

â‡’

Coordinates of any point of line should be in the form of

We know, the intersection point of line and plane lies on the plane, using this,

â‡’

â‡’

â‡’

Therefore, coordinates of point is given by, putting  ,

â‡’ (2, -1, 2)

Therefore, now distance between (-1, -5, -10) and (2, -1, 2) is,

â‡’

â‡’  â‡’ 13 units

### and

Solution:

Given equation of line is

â‡’

Coordinates of any point of line should be in the form of

We know, the intersection point of line and plane lies on the plane, using this,

â‡’ .

â‡’

â‡’

Therefore, coordinates of point is given by, putting ,

â‡’ (14, 12, 10).

Therefore, now distance between the points (2, 12, 5) and (14, 12, 10) is,

â‡’

â‡’  â‡’ 13 units

### Question 5: Find the distance of point (-1, -5, -10) from the point of intersection of the joining A(2, -1, 2) and B(5, 3, 4) with the plane x-y+z=5.

Solution:

Equation of line joining the points A(2, -1, 2) and B(5, 3, 4) is

â‡’

Let,

â‡’

Coordinates of any point on the line is given by

We know, The intersection of line and plane lies on the plane, so,

â‡’

â‡’

Therefore, the coordinates of points is, putting

â‡’ (2, -1, 2)

Now, the distance between the points (-1, -5, -10) and (2, -1, 2) is,

â‡’

â‡’  â‡’ 13 units

### Question 6: Find the distance of point (3, 4, 4) from the point, where the line joining the points A(3, -4, -5) and B(2, -3, 1) intersects the 2x+y+z=7.

Solution:

Equation of line passing through A(3, -4, -5) and B(2, -3, 1) is given by

â‡’

Let

â‡’

Coordinates of any point on the line is given by

We know, The intersection of line and plane lies on the plane, so,

â‡’

â‡’

â‡’

Therefore, the coordinates of points is, putting

â‡’ (1, -2, 7)

Now, the distance between (3, 4, 4) and (1, -2, 7) is,

â‡’

â‡’  = 7 units

### Question 7: Find the distance of point (1, -5, 9) from the plane x- y+ z=5 measured along the line x=y=z.

Solution:

Given, The equation of line is x=y=z, it can also be written as,

, where (1, 1, 1) are direction ratios of the line.

Here we have to measure the distance along the line, the equation of line parallel to x=y=z have same direction ratios (1, 1, 1),

So, the equation of line passing through (1, -5, 9) and having direction ratios (1, 1, 1) is,

â‡’

Let

Coordinates of any point on the line is given by

We know, The intersection of line and plane lies on the plane, so,

â‡’

â‡’

Therefore, the coordinates of point is given by, putting  = (-9, -15, -1)

Now, distance between the points (1, -5, 9) and (-9, -15, -1) is,

â‡’

â‡’  units.

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