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The abscissa of the two points A and B are the roots of the equation `x^2+2a x-b^2=0`and their ordinates are the roots of the equation `x^2+2p x-q^2=0.`Find the equation of the circle with AB as diameter. Also, find itsradius. |
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Answer» Correct Answer - `x^(2)+y^(2)+2ax +2py-(b^(2)+q^(2)) = 0`. Let`(x_(1), y_(1)) and (x_(2), y_(2)) ` be the coordinates of points A and B, respectively. It is given that `x_(1)+x_(2) " are the roots of " x^(2)+2ax-b^(2)=0` `rArr" " x_(1)+x_(2)=-2a and x_(1)x_(2)=-b^(2)" "...(i)` Also, `y_(1) and y_(2) " are the roots of "y^(2)+2py-q^(2)=0` `rArr y_(1)+y_(2)=-2pand y_(1)y_(2)=-q^(2)" "...(ii)` `therefore` The equation of circle with AB as diameter is, `(x-x_(1))(x-x_(2))+(y-y_(1))(y-y_(2))=0` `rArrx^(2)+y^(2)-(x_(1)+x_(2))x-(y_(1)+y_(2))y+(x_(1)x_(2)+y_(1)y_(2))=0` `rArr x^(2) + y^(2)+2ax+2py-(b^(2)+q^(2))=0` and radius `=sqrt(a^(2)+p^(2)+b^(2)+q^(2))` |
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