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NCERT Class XII Mathematics Chapter - - Solutions

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Question 35 - 38: In the following cases, find the coordinates of the foot of theperpendicular drawn from the origin.
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Question : 36 of 78
Marks: +1, -0
3y + 4z – 6 = 0
Solution:  
Let the coordinates of the foot of the perpendicular P from the origin to the plane be (x1,y1,z1)(x_1, y_1, z_1) .
Then the direction ratios of the line OP are x1,y1,z1x_1, y_1, z_1. Writing the equation of the plane in the normal form, we have 35y+45z=65\frac{3}{5}y + \frac{4}{5}z = \frac{6}{5}
where 35,45\frac{3}{5}, \frac{4}{5} are the direction cosines of OP.
Since direction cosines and direction ratios of the line are proportional, we have x10=y135=z145=k\frac{x_1}{0} = \frac{y_1}{\frac{3}{5}} = \frac{z_1}{\frac{4}{5}} = k i.e., x1=0,y1=3k5,z1=4k5x_1 = 0, y_1 = \frac{3k}{5}, z_1 = \frac{4k}{5}
Substituting these values in the equation of the plane, we get k=65k = \frac{6}{5}
Hence the foot of perpendicular is (0,1825,2425)\left(0, \frac{18}{25}, \frac{24}{25}\right)
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