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Question Number 136781 by bramlexs22 last updated on 25/Mar/21

$$ \\ $$ What is the equation of the circle passing though (-2,-4) and (4,6) its center lies on the line 3x- 2y+20=0\\n

Answered by EDWIN88 last updated on 26/Mar/21

Let P(−2,−4) and Q(4,6) , and midpoint PQ is T(((4−2)/2), ((6−4)/2))=T(1,1). let C(a,b) be a center point the circle. We know that vector PQ ⊥ CT ; then { ((PQ = (4−(−2),(6−(−4))=(6,10))),((CT= (1−a,1−b))) :} PQ.CT = 0 ; 6(1−a)+10(1−b)= 0 ⇒16−6a−10b= 0 ; 3a+5b = 8...(i) Because the center point the circle lies on the line 3x−2y+20 = 0 , we get ⇒ 3a−2b = −20...(ii) Now we can solving for a and b , gives 3a+5b = 8 3a −2b = −20 7b = 28 { ((b=4)),((a= −4)) :} , now we want to compute the radius = ∣CP∣ = (√((−2+4)^2 +(−4−4)^2 )) radius = (√(4+64)) = (√(68.)) Finally the equation of the circle is equal to (x+4)^2 +(y−4)^2 = 68 or x^2 +y^2 +8x−8y−34 = 0

$$\mathrm{Let}\:\mathrm{P}\left(−\mathrm{2},−\mathrm{4}\right)\:\mathrm{and}\:\mathrm{Q}\left(\mathrm{4},\mathrm{6}\right)\:,\:\mathrm{and}\:\mathrm{midpoint} \\ $$ $$\mathrm{PQ}\:\mathrm{is}\:\mathrm{T}\left(\frac{\mathrm{4}−\mathrm{2}}{\mathrm{2}},\:\frac{\mathrm{6}−\mathrm{4}}{\mathrm{2}}\right)=\mathrm{T}\left(\mathrm{1},\mathrm{1}\right).\: \\ $$ $$\mathrm{let}\:\mathrm{C}\left(\mathrm{a},\mathrm{b}\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{center}\:\mathrm{point}\:\mathrm{the}\:\mathrm{circle}. \\ $$ $$\mathrm{We}\:\mathrm{know}\:\mathrm{that}\:\mathrm{vector}\:\boldsymbol{\mathrm{PQ}}\:\bot\:\boldsymbol{\mathrm{CT}}\:\:;\:\mathrm{then} \\ $$ $$\begin{cases}{\boldsymbol{\mathrm{\color{mathred}{P}\color{mathred}{Q}}}\color{mathred}{\:}\color{mathred}{=}\color{mathred}{\:}\color{mathred}{\left(}\mathrm{\color{mathred}{4}}\color{mathred}{−}\color{mathred}{\left(}\color{mathred}{−}\mathrm{\color{mathred}{2}}\color{mathred}{\right)}\color{mathred}{,}\color{mathred}{\left(}\mathrm{\color{mathred}{6}}\color{mathred}{−}\color{mathred}{\left(}\color{mathred}{−}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}\color{mathred}{\right)}\color{mathred}{=}\color{mathred}{\left(}\mathrm{\color{mathred}{6}}\color{mathred}{,}\mathrm{\color{mathred}{1}\color{mathred}{0}}\color{mathred}{\right)}\right.}\\{\boldsymbol{\mathrm{\color{mathred}{C}\color{mathred}{T}}}\color{mathred}{=}\color{mathred}{\:}\color{mathred}{\left(}\mathrm{\color{mathred}{1}}\color{mathred}{−}\mathrm{\color{mathred}{a}}\color{mathred}{,}\mathrm{\color{mathred}{1}}\color{mathred}{−}\mathrm{\color{mathred}{b}}\color{mathred}{\right)}}\end{cases} \\ $$ $$\boldsymbol{\mathrm{\color{mathred}{P}\color{mathred}{Q}}}\color{mathred}{.}\boldsymbol{\mathrm{\color{mathred}{C}\color{mathred}{T}}}\color{mathred}{\:}\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{0}}\color{mathred}{\:}\color{mathred}{;}\color{mathbrown}{ }\color{mathred}{\:}\mathrm{\color{mathred}{6}}\color{mathred}{\left(}\mathrm{\color{mathred}{1}}\color{mathred}{−}\mathrm{\color{mathred}{a}}\color{mathred}{\right)}\color{mathred}{+}\mathrm{\color{mathred}{1}\color{mathred}{0}}\color{mathred}{\left(}\mathrm{\color{mathred}{1}}\color{mathred}{−}\mathrm{\color{mathred}{b}}\color{mathred}{\right)}\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{0}} \\ $$ $$\color{mathred}{\Rightarrow}\mathrm{\color{mathblue}{1}\color{mathblue}{6}}\color{mathblue}{−}\mathrm{\color{mathblue}{6}\color{mathblue}{a}}\color{mathblue}{−}\mathrm{\color{mathblue}{1}\color{mathblue}{0}\color{mathblue}{b}}\color{mathblue}{=}\color{mathblue}{\:}\mathrm{\color{mathblue}{0}}\color{mathblue}{\:}\color{mathblue}{;}\color{mathblue}{\:}\mathrm{\color{mathblue}{3}\color{mathblue}{a}}\color{mathblue}{+}\mathrm{\color{mathblue}{5}\color{mathblue}{b}}\color{mathblue}{\:}\color{mathblue}{=}\color{mathblue}{\:}\mathrm{\color{mathblue}{8}}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{\left(}\mathrm{\color{mathblue}{i}}\color{mathblue}{\right)} \\ $$ $$\mathrm{\color{mathblue}{B}\color{mathblue}{e}\color{mathblue}{c}\color{mathblue}{a}\color{mathblue}{u}\color{mathblue}{s}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{c}\color{mathblue}{e}\color{mathblue}{n}\color{mathblue}{t}\color{mathblue}{e}\color{mathblue}{r}}\color{mathblue}{\:}\mathrm{\color{mathblue}{p}\color{mathblue}{o}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{t}}\color{mathblue}{\:}\mathrm{\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{c}\color{mathblue}{i}\color{mathblue}{r}\color{mathblue}{c}\color{mathblue}{l}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{l}\color{mathblue}{i}\color{mathblue}{e}\color{mathblue}{s}} \\ $$ $$\mathrm{\color{mathblue}{o}\color{mathblue}{n}}\color{mathblue}{\:}\mathrm{\color{mathblue}{t}\color{mathblue}{h}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{l}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{3}\color{mathblue}{x}}\color{mathblue}{−}\mathrm{\color{mathblue}{2}\color{mathblue}{y}}\color{mathblue}{+}\mathrm{\color{mathblue}{2}\color{mathblue}{0}}\color{mathblue}{\:}\color{mathblue}{=}\color{mathblue}{\:}\mathrm{\color{mathblue}{0}}\color{mathblue}{\:}\color{mathblue}{,}\color{mathblue}{\:}\mathrm{\color{mathblue}{w}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{g}\color{mathblue}{e}\color{mathblue}{t}}\color{mathblue}{\:} \\ $$ $$\color{mathblue}{\Rightarrow}\color{mathblue}{\:}\mathrm{\color{mathblue}{3}\color{mathblue}{a}}\color{mathblue}{−}\mathrm{\color{mathblue}{2}\color{mathblue}{b}}\color{mathblue}{\:}\color{mathblue}{=}\color{mathblue}{\:}\color{mathblue}{−}\mathrm{\color{mathblue}{2}\color{mathblue}{0}}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{.}\color{mathblue}{\left(}\mathrm{\color{mathblue}{i}\color{mathblue}{i}}\color{mathblue}{\right)} \\ $$ $$\mathrm{\color{mathblue}{N}\color{mathblue}{o}\color{mathblue}{w}}\color{mathblue}{\:}\mathrm{\color{mathblue}{w}\color{mathblue}{e}}\color{mathblue}{\:}\mathrm{\color{mathblue}{c}\color{mathblue}{a}\color{mathblue}{n}}\color{mathblue}{\:}\mathrm{\color{mathblue}{s}\color{mathblue}{o}\color{mathblue}{l}\color{mathblue}{v}\color{mathblue}{i}\color{mathblue}{n}\color{mathblue}{g}}\color{mathblue}{\:}\mathrm{\color{mathblue}{f}\color{mathblue}{o}\color{mathblue}{r}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}}\color{mathblue}{\:}\mathrm{\color{mathblue}{a}\color{mathblue}{n}\color{mathblue}{d}}\color{mathblue}{\:}\mathrm{\color{mathblue}{b}}\color{mathblue}{\:}\color{mathblue}{,}\color{mathblue}{\:}\mathrm{\color{mathblue}{g}\color{mathblue}{i}\color{mathblue}{v}\color{mathblue}{e}\color{mathblue}{s}} \\ $$ $$\color{mathbrown}{ }\color{mathbrown}{\:}\mathrm{\color{mathbrown}{3}\color{mathbrown}{a}}\color{mathbrown}{+}\mathrm{\color{mathbrown}{5}\color{mathbrown}{b}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{8}} \\ $$ $$\color{mathbrown}{ }\color{mathbrown}{\:}\mathrm{\color{mathbrown}{3}\color{mathbrown}{a}}\color{mathbrown}{\:}\color{mathbrown}{−}\mathrm{\color{mathbrown}{2}\color{mathbrown}{b}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\color{mathbrown}{−}\mathrm{\color{mathbrown}{2}\color{mathbrown}{0}} \\ $$ $$ \\ $$ $$\color{mathbrown}{\:}\color{mathbrown}{ }\color{mathbrown}{\:}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{7}\color{mathbrown}{b}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{2}\color{mathbrown}{8}}\color{mathbrown}{\:}\begin{cases}{\mathrm{\color{mathbrown}{b}}\color{mathbrown}{=}\mathrm{\color{mathbrown}{4}}}\\{\mathrm{\color{mathbrown}{a}}\color{mathbrown}{=}\color{mathbrown}{\:}\color{mathbrown}{−}\mathrm{\color{mathbrown}{4}}}\end{cases}\color{mathbrown}{\:}\color{mathbrown}{,}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{n}\color{mathbrown}{o}\color{mathbrown}{w}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{w}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{w}\color{mathbrown}{a}\color{mathbrown}{n}\color{mathbrown}{t}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{o}} \\ $$ $$\mathrm{\color{mathbrown}{c}\color{mathbrown}{o}\color{mathbrown}{m}\color{mathbrown}{p}\color{mathbrown}{u}\color{mathbrown}{t}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{r}\color{mathbrown}{a}\color{mathbrown}{d}\color{mathbrown}{i}\color{mathbrown}{u}\color{mathbrown}{s}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\color{mathbrown}{\mid}\boldsymbol{\mathrm{\color{mathbrown}{C}\color{mathbrown}{P}}}\color{mathbrown}{\mid}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\sqrt{\color{mathbrown}{\left(}\color{mathbrown}{−}\mathrm{\color{mathbrown}{2}}\color{mathbrown}{+}\mathrm{\color{mathbrown}{4}}\color{mathbrown}{\right)}^{\mathrm{\color{mathbrown}{2}}} \color{mathbrown}{+}\color{mathbrown}{\left(}\color{mathbrown}{−}\mathrm{\color{mathbrown}{4}}\color{mathbrown}{−}\mathrm{\color{mathbrown}{4}}\color{mathbrown}{\right)}^{\mathrm{\color{mathbrown}{2}}} } \\ $$ $$\mathrm{\color{mathbrown}{r}\color{mathbrown}{a}\color{mathbrown}{d}\color{mathbrown}{i}\color{mathbrown}{u}\color{mathbrown}{s}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\sqrt{\mathrm{\color{mathbrown}{4}}\color{mathbrown}{+}\mathrm{\color{mathbrown}{6}\color{mathbrown}{4}}}\color{mathbrown}{\:}\color{mathbrown}{=}\color{mathbrown}{\:}\sqrt{\mathrm{\color{mathbrown}{6}\color{mathbrown}{8}}\color{mathbrown}{.}} \\ $$ $$\mathrm{\color{mathbrown}{F}\color{mathbrown}{i}\color{mathbrown}{n}\color{mathbrown}{a}\color{mathbrown}{l}\color{mathbrown}{l}\color{mathbrown}{y}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{e}\color{mathbrown}{q}\color{mathbrown}{u}\color{mathbrown}{a}\color{mathbrown}{t}\color{mathbrown}{i}\color{mathbrown}{o}\color{mathbrown}{n}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{o}\color{mathbrown}{f}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{h}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{c}\color{mathbrown}{i}\color{mathbrown}{r}\color{mathbrown}{c}\color{mathbrown}{l}\color{mathbrown}{e}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{i}\color{mathbrown}{s}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{e}\color{mathbrown}{q}\color{mathbrown}{u}\color{mathbrown}{a}\color{mathbrown}{l}}\color{mathbrown}{\:}\mathrm{\color{mathbrown}{t}\color{mathbrown}{o}} \\ $$ $$\color{mathred}{ }\color{mathred}{\:}\color{mathred}{\left(}\mathrm{\color{mathred}{x}}\color{mathred}{+}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}^{\mathrm{\color{mathred}{2}}} \color{mathred}{+}\color{mathred}{\left(}\mathrm{\color{mathred}{y}}\color{mathred}{−}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}^{\mathrm{\color{mathred}{2}}} \color{mathred}{\:}\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{6}\color{mathred}{8}}\color{mathred}{\:}\mathrm{\color{mathred}{o}\color{mathred}{r}}\color{mathred}{\:} \\ $$ $$\color{mathred}{ }\color{mathred}{\:}\mathrm{\color{mathred}{x}}^{\mathrm{\color{mathred}{2}}} \color{mathred}{+}\mathrm{\color{mathred}{y}}^{\mathrm{\color{mathred}{2}}} \color{mathred}{+}\mathrm{\color{mathred}{8}\color{mathred}{x}}\color{mathred}{−}\mathrm{\color{mathred}{8}\color{mathred}{y}}\color{mathred}{−}\mathrm{\color{mathred}{3}\color{mathred}{4}}\color{mathred}{\:}\color{mathred}{=}\color{mathred}{\:}\mathrm{\color{mathred}{0}} \\ $$

Commented byEDWIN88 last updated on 26/Mar/21

Answered by mr W last updated on 26/Mar/21

say center of circle is (u,v), radius r 3u−2v+20=0 ...(i) (u+2)^2 +(v+4)^2 =r^2 ...(ii) (u−4)^2 +(v−6)^2 =r^2 ...(iii) (ii)−(iii): 3u+5v−8=0 ...(iv) (iv)−(i): 7v−28=0 ⇒v=4 u=((8−5v)/3)=((8−20)/3)=−4 r^2 =(−4+2)^2 +(4+4)^2 =68 eqn. of circle: (x+4)^2 +(y−4)^2 =68

$${say}\:{center}\:{of}\:{circle}\:{is}\:\left({u},{v}\right),\:{radius}\:{r} \\ $$ $$\mathrm{3}{u}−\mathrm{2}{v}+\mathrm{20}=\mathrm{0}\:\:\:...\left({i}\right) \\ $$ $$\left({u}+\mathrm{2}\right)^{\mathrm{2}} +\left({v}+\mathrm{4}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} \:\:\:...\left({ii}\right) \\ $$ $$\left({u}−\mathrm{4}\right)^{\mathrm{2}} +\left({v}−\mathrm{6}\right)^{\mathrm{2}} ={r}^{\mathrm{2}} \:\:\:...\left({iii}\right) \\ $$ $$\left({ii}\right)−\left({iii}\right): \\ $$ $$\mathrm{3}{u}+\mathrm{5}{v}−\mathrm{8}=\mathrm{0}\:\:\:...\left({iv}\right) \\ $$ $$\left({iv}\right)−\left({i}\right): \\ $$ $$\mathrm{7}{v}−\mathrm{28}=\mathrm{0}\:\Rightarrow{v}=\mathrm{4} \\ $$ $${u}=\frac{\mathrm{8}−\mathrm{5}{v}}{\mathrm{3}}=\frac{\mathrm{8}−\mathrm{20}}{\mathrm{3}}=−\mathrm{4} \\ $$ $${r}^{\mathrm{2}} =\left(−\mathrm{4}+\mathrm{2}\right)^{\mathrm{2}} +\left(\mathrm{4}+\mathrm{4}\right)^{\mathrm{2}} =\mathrm{68} \\ $$ $${eqn}.\:{of}\:{circle}: \\ $$ $$\color{mathred}{\left(}{\color{mathred}{x}}\color{mathred}{+}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}^{\mathrm{\color{mathred}{2}}} \color{mathred}{+}\color{mathred}{\left(}{\color{mathred}{y}}\color{mathred}{−}\mathrm{\color{mathred}{4}}\color{mathred}{\right)}^{\mathrm{\color{mathred}{2}}} \color{mathred}{=}\mathrm{\color{mathred}{6}\color{mathred}{8}} \\ $$

Commented byotchereabdullai@gmail.com last updated on 27/Mar/21

nice!

$$\mathrm{nice}! \\ $$

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