Category: Coordinate Geometry

Question Number 118868 by bemath last updated on 20/Oct/20 $${If}\:{the}\:{graphs}\:{of}\:{y}={x}^{\mathrm{2}} +\mathrm{2}{ax}+\mathrm{6}{b}\: \\ $$$${and}\:{y}={x}^{\mathrm{2}} +\mathrm{2}{bx}+\mathrm{6}{a}\:{intersect}\:{at}? \\ $$$${only}\:{one}\:{point}\:{in}\:{the}\:{xy}−{plane} \\ $$$$,\:{what}\:{is}\:{the}\:{x}−{coordinate}\:{of}\:{the} \\ $$$${point}\:{of}\:{intersection}\:? \\ $$ Answered by bobhans…

Question Number 184349 by alcohol last updated on 05/Jan/23 $${I}_{{n}} \:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx} \\ $$$${Relate}\:{I}_{{n}} \:{and}\:{I}_{{n}−\mathrm{1}} \\ $$$${Find}\:{I}_{{n}} \:{in}\:{terms}\:{of}\:{n} \\ $$$${hence}\:{deduce}\:{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}}…

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Question Number 52731 by ajfour last updated on 12/Jan/19 Commented by MJS last updated on 12/Jan/19 $$\mathrm{for}\:{b}={a}\:\left(\mathrm{circle}\right)\:\mathrm{the}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{is} \\ $$$$\mathrm{the}\:\mathrm{one}\:\mathrm{with}\:\mathrm{minimum}\:\mathrm{perimeter}. \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{draw}\:\mathrm{an}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{for}\:\mathrm{a} \\ $$$$\mathrm{circle}\:\mathrm{with}\:{r}={a}\:\mathrm{and}\:\mathrm{then}\:\mathrm{compress}\:\mathrm{with} \\ $$$$\mathrm{the}\:\mathrm{factor}\:\frac{{a}}{{b}}\:\left[{P}=\begin{pmatrix}{{x}}\\{{y}}\end{pmatrix}\:\rightarrow\:{P}'=\begin{pmatrix}{{x}}\\{\frac{{b}}{{a}}{y}}\end{pmatrix}\right]…

Question Number 52627 by ajfour last updated on 10/Jan/19 Commented by ajfour last updated on 10/Jan/19 $${Find}\:{radius}\:{of}\:{such}\:{a}\:{circle}. \\ $$ Answered by mr W last updated…