NCERT Class XI Mathematics - Introduction to Three Dimensional Geometry - Solutions
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Question : 20
Total: 20
If A and B be the points (3, 4, 5) and (–1, 3, –7), respectively, find the equation of the set of points P such that P A 2 + P B 2 k 2
Solution:
Let P(x, y, z) be any point
Then PA =√ ( x − 3 ) 2 + ( y − 4 ) 2 + ( z − 5 ) 2
=√ x 2 + 9 − 6 x + y 2 + 16 − 8 y + z 2 + 25 − 10 z
PB =√ ( x + 1 ) 2 + ( y − 3 ) 2 + ( z + 7 ) 2
=√ x 2 + 1 + 2 x + y 2 + 9 − 6 y + z 2 + 49 + 14 z
Now,P A 2 + P B 2 k 2
∴[ √ x 2 + 9 − 6 x + y 2 + 16 − 8 y + z 2 + 25 − 10 z ] 2 [ √ x 2 + 1 + 2 x + y 2 + 9 − 6 y + z 2 + 49 + 14 z ] 2 k 2
∴x 2 y 2 z 2 x 2 y 2 z 2 k 2
⇒2 x 2 + 2 y 2 + 2 z 2 k 2
⇒ 2( x 2 + y 2 + z 2 − 2 z − 7 y + 2 z ) k 2
⇒x 2 + y 2 + z 2
Then PA =
=
PB =
=
Now,
∴
∴
⇒
⇒ 2
⇒
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