Choose the correct option :
- The planes intersect in a line if-
none of these.
Solution.
The equation of any plane passing through the line of intersection of two planes can be written as
Comparing and we get,
From we get,
So, option (c) is correct.
2. The point in which the line meets the plane is-
Solution.
Let the straight line meets the plane at the point
So, the required point is
So, option (a) is correct.
3. The coordinates of the point where the line joining the points and meets the xy plane are-
(a) (b) (c) (d) none of these
Solution.
Let
The equation of the straight line passing through and is given by
Let the straight line (1) intersects the plane xy at the point
So, from (1) and (2) we get,
So, the point is
Hence, option (b) is correct.
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4. The line joining the points and meets the plane is-
(a) (b) (c) (d)
Solution.
The equation of the straight line passing through the points and is given by
So, the point lies on the plane.
So, the required point is
So, option (b) is correct.
5. A plane meets the coordinate axes in A, B,C such that the centroid of the triangle ABC is the point Then the equation of the plane where is –
(a) (b) (c) (d)
Solution.
If is the centroid of the triangle ABC, then the equation of the plane is
So, option (d) is correct.
6. State which of the following statement is true ?
(a) The plane through the points whose coordinates are and respectively, passes through the point for all values of
(b) The equation of the plane which is parallel to the plane and passing through the point is
(c) The equation of the line of intersection of planes and is
(d) The equation of the straight line and the plane are perpendicular to each other.
Solution.
Option (a) is correct.
Explanation.
The equation of the plane through the points and is
Clearly, the point satisfies the equation of the plane (1).
7. The equation of the plane passing through the point having as the direction ratios of the normal to the plane is-
(a) (b) (c) (d) none of these
Solution.
The equation of the plane can be written as
where the values of are respectively and
So, by we get,
So, option (c) is correct.
8. The value of for which the straight line is parallel to the plane is – (a) (b) (c) (d)
Solution.
The direction ratios of are and the direction ratios of are
The vector equation of the plane perpendicular to and is given by
Now, is perpendicular to
So, option (a) is correct.
9. The intercept made by the plane on the x-axis is – (a) (b) (c) (d)
Solution.
Let
So, the required intercept is given by
So, option (a) is correct.
10. A unit vector parallel to the intersection of planes and is-
(a) (b) (c) (d)
Solution. Let the planes and be denoted by and respectively.
Hence, the required unit vector is given by
So, option (c) is correct.
11. The line will not meet the plane if-
(a) (b)
(c) (d)
Solution.
If then because the line will not meet the plane
So, option (c) is correct.
12. The ratio in which the plane divides the line joining the points and is-
(a) (b) (c) (d)
Solution.
Let the given plane divides the line joining the points and in the ratio
By our supposition,
So, from (1) we get,
Hence, option (d) is correct.
13. The lines and are coplanar if-
(a) (b) (c) (d)
Solution.
The given lines are coplanar if
So, option (b) is correct.
14. The plane , which passes through the point and the line is
(a) (b) (c) (d)
Solution.
The equation of the plane passing through the point is
Since the point lies on the plane (1), so
Since the direction ratios of and are perpendicular to each other , we get
Hence, from (1), (2) and (3) we get,
Hence, option (a) is correct.