Question Number 16067 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{d},\:{d}'\:\mathrm{be}\:\mathrm{two}\:\mathrm{nonparallel}\:\mathrm{lines}\:\mathrm{in}\:\mathrm{the} \\ $$$$\mathrm{plane}\:\mathrm{and}\:\mathrm{let}\:{k}\:>\:\mathrm{0}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{locus}\:\mathrm{of} \\ $$$$\mathrm{points},\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{whose}\:\mathrm{distances}\:\mathrm{to} \\ $$$${d}\:\mathrm{and}\:{d}'\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to}\:{k}. \\ $$ Commented by mrW1 last updated on 29/Jun/17…
Question Number 16064 by Tinkutara last updated on 21/Jun/17 $$\mathrm{Let}\:{ABCD}\:\mathrm{be}\:\mathrm{a}\:\mathrm{convex}\:\mathrm{quadrilateral} \\ $$$$\mathrm{and}\:\mathrm{M}\:\mathrm{a}\:\mathrm{point}\:\mathrm{in}\:\mathrm{its}\:\mathrm{interior}\:\mathrm{such}\:\mathrm{that} \\ $$$$\left[{MAB}\right]\:=\:\left[{MBC}\right]\:=\:\left[{MCD}\right]\:=\:\left[{MDA}\right]. \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{diagonals}\:\mathrm{of} \\ $$$${ABCD}\:\mathrm{passes}\:\mathrm{through}\:\mathrm{the}\:\mathrm{midpoint}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{other}\:\mathrm{diagonal}. \\ $$ Commented by Tinkutara…
Question Number 16053 by vpawarksp@gmail.com last updated on 17/Jun/17 $${number}\:{of}\:{positive}\:{integers}\:\boldsymbol{{a}}\:\boldsymbol{{and}}\:\boldsymbol{{b}}\:\boldsymbol{{and}}\:\boldsymbol{{c}}\:\boldsymbol{{satisfying}} \\ $$$$\boldsymbol{{a}}^{\boldsymbol{{b}}^{\boldsymbol{{c}}} } \boldsymbol{{b}}^{\boldsymbol{{c}}^{\boldsymbol{{a}}} } \boldsymbol{{c}}^{\boldsymbol{{a}}^{\boldsymbol{{b}}} } =\mathrm{5}\boldsymbol{{abc}} \\ $$ Commented by prakash jain last…
Question Number 16014 by chux last updated on 16/Jun/17 $$\mathrm{A}\:\mathrm{cirlce}\:\mathrm{is}\:\mathrm{drawn}\:\mathrm{to}\:\mathrm{touch}\:\mathrm{the} \\ $$$$\mathrm{sides}\:\mathrm{of}\:\mathrm{a}\:\mathrm{triangle}\:\mathrm{whose}\:\mathrm{sides}\:\mathrm{are} \\ $$$$\mathrm{12cm},\mathrm{10cm},\mathrm{and}\:\mathrm{9cm}.\:\mathrm{Find}\:\mathrm{the}\: \\ $$$$\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}. \\ $$ Answered by RasheedSoomro last updated on 16/Jun/17…
Question Number 15987 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Jun/17 Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Jun/17 $${H},{is}\:{a}\:{point}\:{on}\:{circle}\:{with}\:{centerpoint}:{A}. \\ $$$$\:\:{prove}\:\:{that}: \\ $$$$\:\:\:\:\:\:\:\:\angle\:{DHF}=\mathrm{135}^{\bullet} \:. \\ $$ Answered…
Question Number 15982 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Jun/17 Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 17/Jun/17 $${in}\:{triangle}\:{ABC}: \\ $$$$\angle{ABC}=\mathrm{120}^{\bullet} ,{AB}={c},{BC}={a},{CA}={b} \\ $$$${D},{E},{F}:\:{intersects}\:{of}\:\left({in}\right)\:{bisector}\:{of} \\ $$$${angles}\:{with}\:{opposite}\:{sides}. \\…
Question Number 15969 by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Jun/17 Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 16/Jun/17 $$\mathrm{3}\:{circles}\:{have}\:{the}\:{same}\:{center}\:{point}. \\ $$$${there}\:{is}\:{a}\:{equilateral}\:{triangle}\:,{such}\:{that} \\ $$$${any}\:{vertex}\:{located}\:{on}\:{one}\:{circle}\:{as}\:{showen}. \\ $$$$\left.\mathrm{1}\right){how}\:{can}\:{we}\:{draw}\:{such}\:{triangle}? \\ $$$$\left.\mathrm{2}\right){find}\:{area}\:{of}\:{this}\:{triangle}\:{in}\:{terms}\:{of}…
Question Number 147009 by gsk2684 last updated on 17/Jul/21 $${a}\:{parabola}\:{y}={x}^{\mathrm{2}} −\mathrm{15}{x}+\mathrm{36}\:{cuts}\:{the}\: \\ $$$${x}\:{axis}\:{at}\:{P}\:\:{and}\:{Q}.\:{a}\:{circle}\:{is}\:{drawn} \\ $$$${through}\:{P}\:{and}\:{Q}\:{so}\:{that}\:{the}\:{origin} \\ $$$${is}\:{outside}\:{it}.\:{then}\:{find}\:{the}\:{length}\: \\ $$$${of}\:{tangent}\:{to}\:{the}\:{circle}\:{from}\:\left(\mathrm{0},\mathrm{0}\right)? \\ $$ Answered by mr W…
Question Number 147010 by gsk2684 last updated on 17/Jul/21 $${if}\:{the}\:{radius}\:{of}\:{a}\:{circle}\:{touching}\: \\ $$$${parabola}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:\:{at}\:\left(\mathrm{4},\mathrm{4}\right){and}\:{having} \\ $$$${directrix}\:{of}\:{y}^{\mathrm{2}} =\mathrm{4}{x}\:{as}\:{its}\:{normal}\: \\ $$$${is}\:{r},\:{then}\:{find}\:\left[{r}\right]? \\ $$$$\left({where}\:\left[{x}\right]\:{denote}\:{greatest}\:{integer}\:\right. \\ $$$$\left.{lessthan}\:{or}\:{equal}\:{to}\:{x}\right) \\ $$ Commented…
Question Number 81467 by ajfour last updated on 13/Feb/20 Commented by ajfour last updated on 13/Feb/20 $${If}\:{the}\:{circular}\:{area}\:{cut}\:{out}\:{is} \\ $$$${half}\:{the}\:{sector}\:{area},\:{find}\:\alpha. \\ $$ Answered by ajfour last…