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If a variable place in 3-0 space moves in such a way that the sum of its intercepts on the x and y-axis exceeds, the reciprocal of its intercept on the Z-axis is by 2, then all such plane will pass through the point :

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A variable plane moves so t...

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Asked on December 20, 2019 by

Debayani Bose

A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant λ. It passes through a fixed point, which has coordinates

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ANSWER

Let the equation of the variable plane be

a

x

+

b

y

+

c

z

=1 ...(1)

The intercepts on the coordinate axes are a,b,c.

The sum of reciprocals of intercepts in constant λ, therefore

a

1

+

b

1

+

c

1

=λ⇒

a

(1/λ)

+

b

(1/λ)

+

c

(1/λ)

=λ

∴(

λ

1

,

λ

1

,

λ

1

) lies on the plane (1)

Hence, the variable plane (1) always passes through the fixed point (

λ

1

,

λ

1

,

λ

1

)