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A-circle-x-2-y-2-2x-4y-5-0-with-centr-0-is-cut-by-a-line-y-2x-5-at-points-P-and-Q-Show-that-QO-is-perpendicular-to-PO-




Question Number 59287 by pete last updated on 07/May/19
A circle x^2 +y^2 −2x−4y−5=0 with centr  0 is cut by a line y=2x+5 at points P and Q.  Show that QO is perpendicular to PO.
Acirclex2+y22x4y5=0withcentr0iscutbyaliney=2x+5atpointsPandQ.ShowthatQOisperpendiculartoPO.
Answered by tanmay last updated on 07/May/19
solve y=2x+5 and x^2 +y^2 −2x−4y−5=0  x^2 +4x^2 +20x+25−2x−4(2x+5)−5=0  5x^2 +10x=0  x(x+2)=0  when x=0   y=5  when x=−2    y=1  P(0,5)   Q(−2,1)  x^2 −2x+1+y^2 −4y+4−10=0  (x−1)^2 +(y−2)^2 =10  centre O(1,2)  m_1 =slope OP=((5−2)/(0−1))=−3  m_2 =slope OQ=((2−1)/(1−(−2)))=(1/3)  m_1 ×m_2 =−3×(1/3)=−1  so OP⊥OQ
solvey=2x+5andx2+y22x4y5=0x2+4x2+20x+252x4(2x+5)5=05x2+10x=0x(x+2)=0whenx=0y=5whenx=2y=1P(0,5)Q(2,1)x22x+1+y24y+410=0(x1)2+(y2)2=10centreO(1,2)m1=slopeOP=5201=3m2=slopeOQ=211(2)=13m1×m2=3×13=1soOPOQ
Commented by pete last updated on 07/May/19
thanks very much sir
thanksverymuchsir
Commented by tanmay last updated on 07/May/19
most welcome...
mostwelcome

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