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
⇒
Let,
, where is a constant. ⇒
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, ⇒
⇒ Therefore, coordinates of points is given by, putting
we get, ⇒
(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
⇒
Let,
,where is a constant. ⇒
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, ⇒
⇒ Therefore, the coordinates of point is given by, putting
we get, ⇒
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
⇒
Let
where is constant. ⇒
The coordinates of any point on the line is given by
The line crosses the plane, therefore, point must satisfy the plane equation.
⇒
Therefore, The coordinates of point are given by, putting
, ⇒
⇒ (1, -2, 7)
Question 3: Find the distance of the point (-1, -5, -10) from the point of intersection of line
Solution:
Given equation of line is
⇒
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,
⇒
⇒
⇒
Therefore, coordinates of point is given by, putting
, ⇒ (2, -1, 2)
Therefore, now distance between (-1, -5, -10) and (2, -1, 2) is,
⇒
⇒
⇒ 13 units
Question 4: Find the distance of point (2, 12, 5) from the point of intersection of line
Solution:
Given equation of line is
⇒
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,
⇒
. ⇒
⇒
Therefore, coordinates of point is given by, putting
, ⇒ (14, 12, 10).
Therefore, now distance between the points (2, 12, 5) and (14, 12, 10) is,
⇒
⇒
⇒ 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
⇒
Let,
⇒
Coordinates of any point on the line is given by
We know, The intersection of line and plane lies on the plane, so,
⇒
⇒
Therefore, the coordinates of points is, putting
⇒ (2, -1, 2)
Now, the distance between the points (-1, -5, -10) and (2, -1, 2) is,
⇒
⇒
⇒ 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
⇒
Let
⇒
Coordinates of any point on the line is given by
We know, The intersection of line and plane lies on the plane, so,
⇒
⇒
⇒
Therefore, the coordinates of points is, putting
⇒ (1, -2, 7)
Now, the distance between (3, 4, 4) and (1, -2, 7) is,
⇒
⇒
= 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,
⇒
Let
Coordinates of any point on the line is given by
We know, The intersection of line and plane lies on the plane, so,
⇒
⇒
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,
⇒
⇒
units.