NCERT Class 12-Mathematics: Chapter – 11 Three Dimensional Geometry Part 4 (For CBSE, ICSE, IAS, NET, NRA 2023)
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Question 11:
Find the co-ordinates of the foot of perpendicular drawn from the point to the line joining the points and .
Answer:
Let L be the foot of perpendicular drawn from the points to the line passing through B and C as shown in the Fig. 11.2. The equation of line BC by using formula , the equation of the line BC is
Comparing both sides, we get
Thus, the co-ordinate of L are ,
so that the direction ratios of the line AL are , i.e..
Since AL is perpendicular to BC, we have,
The required point is obtained by substituting the value of , in , which is
Question 12:
Find the image of the point in the line
Answer:
Let be the given point and let L be the foot of perpendicular from P to the given line.
The coordinates of a general point on the given line are
If the coordinates of L are , then the direction ratios of PL are .
But the direction ratios of given line which is perpendicular to PL are . Therefore, , which gives . Hence coordinates of L are .
Let be the image of in the given line. Then L is the mid-point of PQ. Therefore,
Hence, the image of in the given line is .
Question 13:
Find the image of the point having position vector in the plane
Answer:
Let the given point be P and Q be the image of P in the plane as shown in the Fig. 11.4.
Then PQ is the normal to the plane. Since PQ passes through P and is normal to the given plane, so the equation of PQ is given by
Since Q lies on the line PQ, the position vector of Q can be expressed as
Again, since R lies on the plane we have
Hence, the position vector of Q is .