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Three straight lines mutually perpendicu...

Three straight lines mutually perpendicular to each other meet in a point P and one of them intersects the x-axis and another intersects the y-axis, while the third line passes through a fixed point(0,0,c) on the z-axis. Then the locus of P is

A

`x^(2)+y^(2)+z^(2)-2cx=0`

B

`x^(2)+y^(2)+z^(2)-2cy=0`

C

`x^(2)+y^(2)+z^(2)-2cz=0`

D

`x^(2)+y^(2)+z^(2)-2c(x+y+z)=0`

Text Solution

Verified by Experts

The correct Answer is:
C
Let point P be (p,q,r) from which three mutually perpendicular liens `L_(1),L_(2),L_(3)` are drawn.
Let line `L_(1)` cut the axis at A(a,0,0), line `L_(2)` meet the y-axis at `B(0,b,0)` and it is given that line `L_(3)` passes through the fixed point C(0,0,c).
`therefore` Directions ratios of `L_(1),L_(2)` and `L_(3)` are (p-a,q,r),
Since lines are mutually perpendicular,
`p(p-a)+q(q-b)+r^(2)=0`...........(1)
`p^(2)+q(q-b)+r(r-c)=0` ...........(2)
and `p(p-a)+q^(2)+r(r-c)=0`...........(3)
From Eq. (2) + Eq. (3) - Eq.(1), we get
`p^(2)+q^(2)+2r(r-c)-r^(2)=0`
`therefore` Locus is `x^(2)+y^(2)+z^(2)-2cz=0`
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