If-p-x-1-x-2-x-i-and-q-y-1-y-2-y-i-form-two-infinite-arithmetic-sequences-with-common-difference-a-and-b-respectively-then-find-the-locus-of-the-point-where-1-n-

Question Number 61274 by alphaprime last updated on 31/May/19

If p , x_1 ,x_2 ,...x_i and q,y_1 ,y_2 ,...y_i form two infinite arithmetic sequences with common difference a and b respectively , then find the locus of the point ( α , β ) where α = (1/n) Σ_(i=1) ^n x_i and β= (1/n) Σ_(i=1) ^n y_(i .)

$${If}\:{p}\:,\:{x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,…{x}_{{i}} \:{and}\:{q},{y}_{\mathrm{1}} ,{y}_{\mathrm{2}} ,…{y}_{{i}} \:{form}\:{two}\: \\ $$$${infinite}\:{arithmetic}\:{sequences}\:{with}\:{common}\: \\ $$$${difference}\:\:{a}\:{and}\:{b}\:{respectively}\:, \\ $$$${then}\:{find}\:{the}\:{locus}\:{of}\:{the}\:{point}\:\left(\:\alpha\:,\:\beta\:\right)\: \\ $$$${where}\:\alpha\:=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}} {x}_{{i}} \:{and}\:\beta=\:\frac{\mathrm{1}}{{n}}\:\sum_{{i}=\mathrm{1}} ^{{n}} {y}_{{i}\:.} \\ $$

Answered by tanmay last updated on 31/May/19

α=((x_1 +x_2 +x_3 +...+x_n )/n) α=(((p+a)+(p+2a)+...+(p+na))/n) α=((np+a[((n(n+1))/2)])/n) α=p+((a(n+1))/2) β=q+((b(n+1))/2) ((α−p)/(a/2))=((β−q)/(b/2)) so locus... b(x−p)=a(y−q) bx−ay+aq−bp=0 ean of straight line→locus st line...

$$\alpha=\frac{{x}_{\mathrm{1}} +{x}_{\mathrm{2}} +{x}_{\mathrm{3}} +…+{x}_{{n}} }{{n}} \\ $$$$\alpha=\frac{\left({p}+{a}\right)+\left({p}+\mathrm{2}{a}\right)+…+\left({p}+{na}\right)}{{n}} \\ $$$$\alpha=\frac{{np}+{a}\left[\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}\right]}{{n}} \\ $$$$\alpha={p}+\frac{{a}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\beta={q}+\frac{{b}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\ $$$$\frac{\alpha−{p}}{\frac{{a}}{\mathrm{2}}}=\frac{\beta−{q}}{\frac{{b}}{\mathrm{2}}} \\ $$$${so}\:{locus}… \\ $$$${b}\left({x}−{p}\right)={a}\left({y}−{q}\right) \\ $$$${bx}−{ay}+{aq}−{bp}=\mathrm{0} \\ $$$${ean}\:{of}\:{straight}\:{line}\rightarrow{locus}\:{st}\:{line}… \\ $$

Commented by bhanukumarb2@gmail.com last updated on 31/May/19