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Category: Coordinate Geometry

Evaluate-0-1-1-2-x-2-y-dx-y-2-x-dy-along-a-straight-line-from-0-1-to-1-2-

Question Number 139924 by EDWIN88 last updated on 02/May/21 $$\:\:\:\:\:\:\mathrm{Evaluate}\:\int_{\left(\mathrm{0},\mathrm{1}\right)} ^{\left(\mathrm{1},\mathrm{2}\right)} \:\left[\:\left(\mathrm{x}^{\mathrm{2}} −\mathrm{y}\right)\mathrm{dx}\:+\:\left(\mathrm{y}^{\mathrm{2}} +\mathrm{x}\right)\:\mathrm{dy}\:\right]\: \\ $$$$\mathrm{along}\:\mathrm{a}\:\mathrm{straight}\:\mathrm{line}\:\mathrm{from}\:\left(\mathrm{0},\mathrm{1}\right)\:\mathrm{to}\:\left(\mathrm{1},\mathrm{2}\right). \\ $$ Answered by TheSupreme last updated on 02/May/21…

Let-consider-I-R-2-a-parametric-curve-1-Prove-that-if-a-lt-b-and-a-b-then-there-exist-t-0-a-b-such-as-t-0-is-colinear-to-b-a-2-Show-that-if-is-regular-and-the-

Question Number 74300 by ~blr237~ last updated on 21/Nov/19 $${Let}\:{consider}\:\:\gamma\:\::{I}\rightarrow\mathbb{R}^{\mathrm{2}} \:\:{a}\:{parametric}\:{curve}\: \\ $$$$\left.\mathrm{1}\left.\right){Prove}\:{that}\:{if}\:\:{a}<{b}\:\:{and}\:\:\gamma\left({a}\right)\neq\gamma\left({b}\right)\:{then}\:{there}\:{exist}\:\:{t}_{\mathrm{0}} \in\right]{a},{b}\left[\:\:\right. \\ $$$${such}\:{as}\:\:\gamma'\left({t}_{\mathrm{0}} \right)\:\:{is}\:{colinear}\:{to}\:\gamma\left({b}\right)−\gamma\left({a}\right)\: \\ $$$$\left.\mathrm{2}\right){Show}\:{that}\:{if}\:\:\gamma\:{is}\:{regular}\:{and}\:{the}\:\:{function}\:{f}\::{I}\rightarrow\mathbb{R}\:\:\:\:{t}\rightarrow{f}\left({t}\right)=\mid\mid\gamma\left({t}\right)−{O}\left(\mathrm{0},\mathrm{0}\right)\:\mid\mid\:\:{is}\:{maximal}\:{in}\:{t}_{\mathrm{0}} \in{I} \\ $$$${Then}\:\:\mid{K}_{\gamma} \left({t}_{\mathrm{0}} \right)\mid\geqslant\frac{\mathrm{1}}{{f}\left({t}_{\mathrm{0}} \right)}…

Prove-that-S-x-y-z-R-3-x-2-y-2-z-2-is-a-surface-and-find-out-if-possible-the-tangent-plan-in-O-0-0-0-

Question Number 74301 by ~blr237~ last updated on 21/Nov/19 $${Prove}\:{that}\:\:{S}=\left\{\left({x},{y},{z}\right)\in\mathbb{R}^{\mathrm{3}} \backslash\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={z}^{\mathrm{2}} \:\right\}\:{is}\:{a}\:{surface}\: \\ $$$${and}\:{find}\:{out}\:{if}\:{possible}\:{the}\:{tangent}\:{plan}\:{in}\:{O}\left(\mathrm{0},\mathrm{0},\mathrm{0}\right). \\ $$ Answered by mind is power last updated…

show-that-every-sphere-through-the-circle-x-2-y-2-2ax-r-2-0-z-0-z-0-cuts-orthogonally-every-sphere-through-the-circle-x-2-z-2-r-2-y-o-

Question Number 8756 by trapti rathaur@ gmail.com last updated on 25/Oct/16 $${show}\:{that}\:{every}\:{sphere}\:{through}\:{the}\:{circle}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} −\mathrm{2}{ax}+{r}^{\mathrm{2}} =\mathrm{0},{z}=\mathrm{0} \\ $$$$,{z}=\mathrm{0}\:\:\:\:\:\:\:{cuts}\:{orthogonally}\:{every}\:{sphere}\:{through}\:{the}\:{circle}\: \\ $$$${x}^{\mathrm{2}} +{z}^{\mathrm{2}} ={r}^{\mathrm{2}} ,\:{y}={o}\:. \\ $$ Terms…

find-the-equation-of-the-sphere-which-touches-the-plane-3x-2y-z-2-0-at-the-point-1-2-1-and-cuts-orthogonally-the-the-sphere-x-2-y-2-z-2-4x-6y-4-0-

Question Number 8755 by trapti rathaur@ gmail.com last updated on 25/Oct/16 $${find}\:{the}\:{equation}\:{of}\:{the}\:{sphere}\:{which}\:{touches}\:{the}\:{plane}\: \\ $$$$\mathrm{3}{x}+\mathrm{2}{y}−{z}+\mathrm{2}=\mathrm{0}\:{at}\:{the}\:{point}\:\left(\mathrm{1},−\mathrm{2},\mathrm{1}\right)\:{and}\:{cuts}\:{orthogonally}\:{the} \\ $$$${the}\:{sphere}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{6}{y}+\mathrm{4}=\mathrm{0} \\ $$ Terms of Service Privacy…

Let-consider-I-R-2-a-parametric-curve-defined-as-t-I-t-t-2-1-t-3-1-2t-t-3-1-Prove-that-for-a-b-c-I-a-b-c-are-on-the-same-lign-iff-abc-a-b-c-1-

Question Number 74280 by ~blr237~ last updated on 21/Nov/19 $${Let}\:\:{consider}\:\alpha\::\:{I}\rightarrow\mathbb{R}^{\mathrm{2}} \:\:{a}\:{parametric}\:{curve}\:{defined}\:{as} \\ $$$$\forall\:{t}\in{I}\:\:\:\alpha\left({t}\right)=\left(\frac{{t}^{\mathrm{2}} −\mathrm{1}}{{t}^{\mathrm{3}} −\mathrm{1}}\:,\frac{\mathrm{2}{t}}{{t}^{\mathrm{3}} −\mathrm{1}}\right)\: \\ $$$${Prove}\:{that}\:{for}\:{a},{b},{c}\in{I}\:\:\: \\ $$$$\:\:\alpha\left({a}\right),\alpha\left({b}\right),\alpha\left({c}\right)\:{are}\:{on}\:{the}\:{same}\:{lign}\:{iff}\:\:{abc}={a}+{b}+{c}+\mathrm{1} \\ $$ Commented by MJS…

if-any-angle-of-equilateral-triangle-is-1-2-and-any-one-side-is-x-3y-5-0-then-equation-of-the-other-two-sides-are-

Question Number 8608 by tawakalitu last updated on 17/Oct/16 $$\mathrm{if}\:\mathrm{any}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{equilateral}\:\mathrm{triangle}\:\mathrm{is} \\ $$$$\left(−\:\mathrm{1},\:\mathrm{2}\right)\:\mathrm{and}\:\mathrm{any}\:\mathrm{one}\:\mathrm{side}\:\mathrm{is}\:\:\mathrm{x}\:−\:\sqrt{\mathrm{3y}}\:+\:\mathrm{5}\:=\:\mathrm{0} \\ $$$$\mathrm{then}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{the}\:\mathrm{other}\:\mathrm{two}\:\mathrm{sides}\:\mathrm{are}\:?? \\ $$ Answered by sandy_suhendra last updated on 18/Oct/16 Answered by…

Question-73919

Question Number 73919 by necxxx last updated on 16/Nov/19 Commented by necxxx last updated on 16/Nov/19 $${Good}\:{day}\:{sirs}.\:{This}\:{question}\:{was}\:{formed} \\ $$$${and}\:{solved}\:{by}\:{some}\:{of}\:{us}\:{here}.\:{I}\:{really}\: \\ $$$${do}\:{not}\:{remember}\:{the}\:{question}\:{or} \\ $$$${approaches}\:{applied}.\:{Please}\:{help}. \\ $$$${Thanks}\:{in}\:{advance}.…

Find-the-equation-of-the-perpendicular-bisector-of-the-line-joining-the-points-5-4-to-the-point-9-3-

Question Number 8243 by lepan last updated on 04/Oct/16 $${Find}\:{the}\:{equation}\:{of}\:{the}\:{perpendicular}\:{bisector}\:{of}\:{the}\:{line}\:{joining}\:{the}\:{points}\:\left(−\mathrm{5},\mathrm{4}\right)\:{to}\:{the}\:{point}\:\left(\mathrm{9},−\mathrm{3}\right) \\ $$$$ \\ $$ Answered by sandy_suhendra last updated on 04/Oct/16 $$\mathrm{let}\:\mathrm{A}\left(−\mathrm{5},\mathrm{4}\right)\:\mathrm{and}\:\mathrm{B}\left(\mathrm{9},−\mathrm{3}\right) \\ $$$$\mathrm{P}\:\mathrm{is}\:\mathrm{midpoint}\:\mathrm{of}\:\mathrm{AB}\:\mathrm{so}\:\mathrm{P}\left(\frac{−\mathrm{5}+\mathrm{9}}{\mathrm{2}}\:,\:\frac{\mathrm{4}−\mathrm{3}}{\mathrm{2}}\:\right)=\mathrm{P}\left(\mathrm{2},\frac{\mathrm{1}}{\mathrm{2}}\right) \\…