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Question Number 17653 by 1kanika# last updated on 09/Jul/17
A line segment moves in the plane with its end points on the coordinate axes so that the sum of the length of its intersect on the coordinate axes is a constant C . Find the locus of the mid points of this segment . Ans. is 8(∣x∣^3 +∣y∣^3 )=C . Λ means power . pls. solve it.
$$\mathrm{A}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{the}\:\mathrm{plane} \\ $$$$\mathrm{with}\:\mathrm{its}\:\mathrm{end}\:\mathrm{points}\:\mathrm{on}\:\mathrm{the}\:\mathrm{coordinate} \\ $$$$\mathrm{axes}\:\mathrm{so}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{length} \\ $$$$\mathrm{of}\:\mathrm{its}\:\mathrm{intersect}\:\mathrm{on}\:\mathrm{the}\:\mathrm{coordinate}\: \\ $$$$\mathrm{axes}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{C}\:. \\ $$$$\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of}\:\mathrm{the}\:\mathrm{mid}\:\mathrm{points}\:\mathrm{of} \\ $$$$\mathrm{this}\:\mathrm{segment}\:. \\ $$$$\mathrm{Ans}.\:\mathrm{is}\:\:\:\mathrm{8}\left(\mid\mathrm{x}\mid^{\mathrm{3}} +\mid\mathrm{y}\mid^{\mathrm{3}} \right)=\mathrm{C}\:. \\ $$$$\Lambda\:\:\mathrm{means}\:\mathrm{power}\:.\:\mathrm{pls}.\:\mathrm{solve}\:\mathrm{it}. \\ $$
Commented by mrW1 last updated on 09/Jul/17
the locus should be a straight line: x+y=(C/2)
$$\mathrm{the}\:\mathrm{locus}\:\mathrm{should}\:\mathrm{be}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}: \\ $$$$\mathrm{x}+\mathrm{y}=\frac{\mathrm{C}}{\mathrm{2}} \\ $$
Commented by mrW1 last updated on 09/Jul/17
let us say intersection at y−axis is (0,b) and intersection at x−axis is (a,0) a+b=constant=C mid point of seqment is (x,y) x=((0+a)/2) y=((0+b)/2) ⇒x+y=((a+b)/2)=(C/2)
$$\mathrm{let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{intersection}\:\mathrm{at}\:\mathrm{y}−\mathrm{axis}\:\mathrm{is} \\ $$$$\left(\mathrm{0},\mathrm{b}\right)\:\mathrm{and}\:\mathrm{intersection}\:\mathrm{at}\:\mathrm{x}−\mathrm{axis}\:\mathrm{is} \\ $$$$\left(\mathrm{a},\mathrm{0}\right) \\ $$$$\mathrm{a}+\mathrm{b}=\mathrm{constant}=\mathrm{C} \\ $$$$\mathrm{mid}\:\mathrm{point}\:\mathrm{of}\:\mathrm{seqment}\:\mathrm{is}\:\left(\mathrm{x},\mathrm{y}\right) \\ $$$$\mathrm{x}=\frac{\mathrm{0}+\mathrm{a}}{\mathrm{2}} \\ $$$$\mathrm{y}=\frac{\mathrm{0}+\mathrm{b}}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{x}+\mathrm{y}=\frac{\mathrm{a}+\mathrm{b}}{\mathrm{2}}=\frac{\mathrm{C}}{\mathrm{2}} \\ $$
Commented by 1kanika# last updated on 09/Jul/17
but the answer is 8(∣x∣Λ3+∣y∣Λ3)=C
$$\mathrm{but}\:\mathrm{the}\:\mathrm{answer}\:\mathrm{is}\:\:\mathrm{8}\left(\mid\mathrm{x}\mid\Lambda\mathrm{3}+\mid\mathrm{y}\mid\Lambda\mathrm{3}\right)=\mathrm{C} \\ $$
Commented by 1kanika# last updated on 09/Jul/17
where Λ means power
$$\mathrm{where}\:\:\Lambda\:\mathrm{means}\:\mathrm{power} \\ $$
Commented by 1kanika# last updated on 09/Jul/17
Locus of mid points is not a straight line as the line segment is moves all around of plane.
$$\mathrm{Locus}\:\mathrm{of}\:\mathrm{mid}\:\mathrm{points}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{straight} \\ $$$$\mathrm{line}\:\mathrm{as}\:\mathrm{the}\:\mathrm{line}\:\mathrm{segment}\:\mathrm{is}\:\mathrm{moves}\:\mathrm{all} \\ $$$$\mathrm{around}\:\mathrm{of}\:\mathrm{plane}. \\ $$
Commented by mrW1 last updated on 10/Jul/17
then please post a picture to show what your question means.
$$\mathrm{then}\:\mathrm{please}\:\mathrm{post}\:\mathrm{a}\:\mathrm{picture}\:\mathrm{to}\:\mathrm{show} \\ $$$$\mathrm{what}\:\mathrm{your}\:\mathrm{question}\:\mathrm{means}. \\ $$

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