NCERT Class 12-Mathematics: Chapter –11 Three Dimensional Geometry Part 4

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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,

Fig.11.2-AL is perpendicular to BC

AL is Perpendicular to BC

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.

Fig-11.3-The foot of perpendicular from P to the given line

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.

Fig-11.4-The image of P in the plane

The Image of P in the Plane

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 .

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