Question and Answers Forum

All Questions      Topic List

Coordinate Geometry Questions

Previous in All Question      Next in All Question      

Previous in Coordinate Geometry      Next in Coordinate Geometry      

Question Number 15868 by tawa tawa last updated on 14/Jun/17

Find the equation and radius of the circumcircle of the triangle formed by the three line: 2y − 9x + 26 = 0 9y + 2x + 32 = 0 11y − 7x − 27 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{and}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circumcircle}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{formed}\:\mathrm{by}\: \\ $$$$\mathrm{the}\:\mathrm{three}\:\mathrm{line}: \\ $$$$\mathrm{2y}\:−\:\mathrm{9x}\:+\:\mathrm{26}\:=\:\mathrm{0} \\ $$$$\mathrm{9y}\:+\:\mathrm{2x}\:+\:\mathrm{32}\:=\:\mathrm{0} \\ $$$$\mathrm{11y}\:−\:\mathrm{7x}\:−\:\mathrm{27}\:=\:\mathrm{0} \\ $$

Answered by Tinkutara last updated on 14/Jun/17

2y − 9x + 26 = 0 ...(i) 9y + 2x + 32 = 0 ...(ii) 11y − 7x − 27 = 0 ...(iii) Solving (i) and (ii), A = (2, −4) Solving (ii) and (iii), C = (−7, −2) Solving (iii) and (i), B = (4, 5) Slope of (i) = (9/2) and that of (ii) = ((−2)/9) so AB ⊥ AC and ΔABC is right angled at A. ∴ Circumcenter = midpoint of BC and circumradius = ((BC)/2) . Circumcenter = (−1.5, 1.5) and circumradius = ((√(170))/2) Equation of circumcircle = (x + 1.5)^2 + (y − 1.5)^2 = ((170)/4) = 42.5

$$\mathrm{2}{y}\:−\:\mathrm{9}{x}\:+\:\mathrm{26}\:=\:\mathrm{0}\:...\left({i}\right) \\ $$$$\mathrm{9}{y}\:+\:\mathrm{2}{x}\:+\:\mathrm{32}\:=\:\mathrm{0}\:...\left({ii}\right) \\ $$$$\mathrm{11}{y}\:−\:\mathrm{7}{x}\:−\:\mathrm{27}\:=\:\mathrm{0}\:...\left({iii}\right) \\ $$$$\mathrm{Solving}\:\left({i}\right)\:\mathrm{and}\:\left({ii}\right),\:{A}\:=\:\left(\mathrm{2},\:−\mathrm{4}\right) \\ $$$$\mathrm{Solving}\:\left({ii}\right)\:\mathrm{and}\:\left({iii}\right),\:{C}\:=\:\left(−\mathrm{7},\:−\mathrm{2}\right) \\ $$$$\mathrm{Solving}\:\left({iii}\right)\:\mathrm{and}\:\left({i}\right),\:{B}\:=\:\left(\mathrm{4},\:\mathrm{5}\right) \\ $$$$\mathrm{Slope}\:\mathrm{of}\:\left({i}\right)\:=\:\frac{\mathrm{9}}{\mathrm{2}}\:\mathrm{and}\:\mathrm{that}\:\mathrm{of}\:\left({ii}\right)\:=\:\frac{−\mathrm{2}}{\mathrm{9}} \\ $$$$\mathrm{so}\:{AB}\:\bot\:{AC}\:\mathrm{and}\:\Delta{ABC}\:\mathrm{is}\:\mathrm{right}\:\mathrm{angled} \\ $$$$\mathrm{at}\:{A}.\:\therefore\:\mathrm{Circumcenter}\:=\:\mathrm{midpoint}\:\mathrm{of}\:{BC} \\ $$$$\mathrm{and}\:\mathrm{circumradius}\:=\:\frac{{BC}}{\mathrm{2}}\:. \\ $$$$\mathrm{Circumcenter}\:=\:\left(−\mathrm{1}.\mathrm{5},\:\mathrm{1}.\mathrm{5}\right)\:\mathrm{and} \\ $$$$\mathrm{circumradius}\:=\:\frac{\sqrt{\mathrm{170}}}{\mathrm{2}} \\ $$$$\mathrm{Equation}\:\mathrm{of}\:\mathrm{circumcircle}\:= \\ $$$$\left({x}\:+\:\mathrm{1}.\mathrm{5}\right)^{\mathrm{2}} \:+\:\left({y}\:−\:\mathrm{1}.\mathrm{5}\right)^{\mathrm{2}} \:=\:\frac{\mathrm{170}}{\mathrm{4}}\:=\:\mathrm{42}.\mathrm{5} \\ $$

Commented by tawa tawa last updated on 14/Jun/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com