Through the point p three mutually perpendicular straight lines are drawn one passes through a fixed point c on Z axis while the others intersect the x axis and y axis respectively. Show that the locus of p is a sphere

Dear Student,
According to the problem

Let point P be (p,q,r) from which three mutually perpendicular lines L1,L2,L3 are drawn.
Let lineL1 cut the axis at A(a,0,0), lineL2 meet the y-axis atB(0,b,0) and it is given that lineL3 passes through the fixed point C(0,0,c).
∴∴Directions ratios ofL1,L2 and L3 are (p-a,q,r),
Since lines are mutually perpendicular,
p(p−a)+q(q−b)+r²=0...........(1)
p²+q(q−b)+r(r−c)=0...........(2)
andp(p−a)+q²+r(r−c)=0...........(3)
From Eq. (2) + Eq. (3) - Eq.(1), we get
p²+q²+2r(r−c)−r²=0
∴∴Locus isx²+y²+z²−2cz=0 which is the equation of sphere.

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Yuvraj Singh