1.

Let A(0,6,8) and B(15,20,0) are two given points and P(lambda, 0, 0) is a point on x-axis such that PA+PB is minimum If perpendicular distance of origin from the plane passing through P,A,B is d then ([.] represent greatest integers function and {.} represent fracional part)

Answer»

<P>`[d]`
`[d]=5`
`[(1)/({d})]=3`
`[(1)/({d})]=5`

Solution :`therefore` Finding minimum value of
`PA+PB= sqrt(alpha^(2)+100)+sqrt((alpha-15)^(2)+400)`
is same as finding minimum value of
P'A' + P'B' where `P'(alpha,0),A'(0,10)` and are `B'(15,-20)`
which is possible only when P', A', B' are collinear.
` implies alpha=5`
`therefore`Equation of plane passing through
`P(5,0,0),A(0,6,8)` and `B(15,20,0)` is
`2x-y+2z=10`
` implies d=(10)/(3)`and `(alpha,beta,lambda)=((40)/(9),(-20)/(9),(40)/(9))`


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