Question Number 216750 by sniper237 last updated on 17/Feb/25 $${If}\:\:{a}−{b}=\sqrt{{ab}}\:\:\:{find}\:\:\:{the}\:{value}\:{of}\:\frac{{a}−{b}}{{a}+{b}} \\ $$ Answered by mr W last updated on 17/Feb/25 $$\left({a}+{b}\right)^{\mathrm{2}} =\left({a}−{b}\right)^{\mathrm{2}} +\mathrm{4}{ab}={ab}+\mathrm{4}{ab}=\mathrm{5}{ab} \\ $$$$\Rightarrow{a}+{b}=\pm\sqrt{\mathrm{5}{ab}}…
Question Number 216695 by sniper237 last updated on 16/Feb/25 $$\int_{\mathrm{0}} ^{\mathrm{2}\pi} \frac{{dx}}{\mathrm{1}+{sinxcosx}}\overset{?} {=}\:\frac{\mathrm{4}\pi{ln}\mathrm{2}}{\:\sqrt{\mathrm{3}}}\:\: \\ $$ Answered by Ghisom last updated on 16/Feb/25 $$\underset{\mathrm{0}} {\overset{\mathrm{2}\pi} {\int}}\frac{{dx}}{\mathrm{1}+\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}}=\mathrm{8}\underset{\pi/\mathrm{4}}…
Question Number 216607 by Tawa11 last updated on 12/Feb/25 Answered by Rasheed.Sindhi last updated on 12/Feb/25 $$\mathrm{4}{x}^{\mathrm{2}} +{bx}−\mathrm{45}=\left({hx}+{k}\right)\left({x}+{j}\right);{h},{k},{j}\in\mathbb{Z} \\ $$$$\:\:\:\:\:\:\:\:={hx}^{\mathrm{2}} +\left({hj}+{k}\right){x}+{kj} \\ $$$${kj}=−\mathrm{45}\Rightarrow{j}=−\frac{\mathrm{45}}{{k}} \\ $$$$\frac{\mathrm{45}}{{k}}=−{j}\:\in\:\mathbb{Z}\:\rightarrow\left({D}\right)…
Question Number 216630 by Tawa11 last updated on 12/Feb/25 the circles x² + y² -4x -2y +3 =0 and x² +y² + 2x +4y -3…
Question Number 216387 by Tawa11 last updated on 06/Feb/25 Answered by mr W last updated on 06/Feb/25 $$\left(\mathrm{6},\:\mathrm{0}\right)\: \\ $$$$\left(\mathrm{6}−\mathrm{6}\right)^{\mathrm{2}} +\left(\mathrm{0}−\mathrm{3}+{a}\right)^{\mathrm{2}} =\mathrm{5}^{\mathrm{2}} \\ $$$$\Rightarrow−\mathrm{3}+{a}=\mathrm{5}\:\Rightarrow{a}=\mathrm{8} \\…
Question Number 216060 by mr W last updated on 26/Jan/25 Commented by mr W last updated on 26/Jan/25 $${find}\:{radius}\:{of}\:{inscribed}\:{circle}\:{r}=? \\ $$ Commented by Tawa11 last…
Question Number 215230 by Mingma last updated on 01/Jan/25 Answered by A5T last updated on 01/Jan/25 Commented by A5T last updated on 01/Jan/25 $$\mathrm{AC}=\mathrm{BC}=\mathrm{a};\:\mathrm{AB}=\mathrm{c};\:\:\angle\mathrm{CAB}=\theta;\:\mathrm{IE}\parallel\mathrm{AB} \\…
Question Number 215004 by efronzo1 last updated on 25/Dec/24 Commented by mr W last updated on 26/Dec/24 $$\left({xy}+\frac{\lambda^{\mathrm{2}} }{{xy}}\right)_{{min}} =\frac{\mathrm{1}}{\mathrm{4}}+\mathrm{4}\lambda^{\mathrm{2}} \\ $$ Commented by efronzo1…
Question Number 214784 by efronzo1 last updated on 19/Dec/24 $$\:\:\:\downharpoonleft\underline{\:} \\ $$ Answered by A5T last updated on 19/Dec/24 $$\sqrt{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }+\sqrt{{c}^{\mathrm{2}} +{d}^{\mathrm{2}} }\geqslant\sqrt{\left({a}+{c}\right)^{\mathrm{2}} +\left({b}+{d}\right)^{\mathrm{2}}…
Question Number 214020 by ajfour last updated on 24/Nov/24 Commented by ajfour last updated on 24/Nov/24 $${Red}\:{area}\:=\:{Blue}\:{area}.\:{Find}\:{r}\:{in} \\ $$$${terms}\:{of}\:{k},\:\phi. \\ $$ Answered by dionigi last…