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A variable plane moves in such a way that the sum of the reciprocals ofits intercepts on the three coordinate axes is constant. Show that the planepasses through a fixed point. |
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Answer» Let the equation of the variable plane be `x/a+y/b+z/c=1`…………….(i) Then, it makes intercepts a,b,c with the coordinate axes. `therefore 1/a+1/b+1/c=k,` where k is a constant (given) `rArr 1/(ka)+1/(kb)+1/(kc)=1 rArr 1/a(1/k)+1/b(1/k)+1/c(1/k)=1`. `rArr (1/k,1/k,1/k)` satisfies (i). Hence, the given plane passes through a fixed point `(1/k,1/k,1/k)`. |
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