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Author: Tinku Tara

Question-218410

Question Number 218410 by Spillover last updated on 09/Apr/25 Answered by Nicholas666 last updated on 09/Apr/25 $$\:{given};{AB},{CD}\:{tangents}\:{to}\:{circle}\:{O}: \\ $$$${M}\:{midpoint}\:{of}\:{BC};\:{AD}\mid\mid{BC};\:{BD}={x},{AB}={y},{CD}={z} \\ $$$${prove}\:{x}^{\mathrm{2}} ={y}.{z} \\ $$$${solution} \\…

Let-gt-0-fixed-Solve-for-real-numbers-the-system-x-2-yz-2-y-2-zx-7-2-z-2-xy-5-2-

Question Number 218364 by hardmath last updated on 08/Apr/25 $$\mathrm{Let}\:\:\:\boldsymbol{\lambda}>\mathrm{0}\:\:\:\mathrm{fixed} \\ $$$$\mathrm{Solve}\:\mathrm{for}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{the}\:\mathrm{system}:\:\:\:\begin{cases}{\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{yz}\:=\:\lambda^{\mathrm{2}} }\\{\mathrm{y}^{\mathrm{2}} \:−\:\mathrm{zx}\:=\:\mathrm{7}\lambda^{\mathrm{2}} }\\{\mathrm{z}^{\mathrm{2}} \:−\:\mathrm{xy}\:=\:−\mathrm{5}\lambda^{\mathrm{2}} }\end{cases} \\ $$ Answered by vnm last updated…

Find-lim-n-k-1-n-i-1-k-i-k-i-1-3-1-

Question Number 218365 by hardmath last updated on 08/Apr/25 $$\mathrm{Find}:\:\:\:\boldsymbol{\Omega}\:=\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\mathrm{lim}}\:\underset{\boldsymbol{\mathrm{k}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{n}}} {\sum}}\:\left[\:\underset{\boldsymbol{\mathrm{i}}=\mathrm{1}} {\overset{\boldsymbol{\mathrm{k}}} {\sum}}\:\mathrm{i}\:\left(\mathrm{k}\:−\:\mathrm{i}\:+\:\frac{\mathrm{1}}{\mathrm{3}}\right)\right]^{−\mathrm{1}} =\:? \\ $$ Answered by vnm last updated on 08/Apr/25…