InterviewSolution
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InterviewSolution
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A(x_(1),y_(1)) and B(x_(2),y_(2)) are any two distinct points on the parabola y=ax^(2) +bx+c.if P(x_(3),y_(3)) be the point on the are AB where the tangent is parallel to the chord AB, then |
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Answer» <P>`x_(3)` is the A.M. between `x_(1)` and `x_(2)` `(x_(3),y_(3))=2ax_(3)+b=(x_(2)-y_(1))/(x_(2)-x_(1))[given]...(1)` `{As the tangent is (y+y_(3))/(2)=ax x_(3) +b((x+x_(3))/(2))+c}` `because` A and B LIE on the parabola, ` THEREFORE y_(1) = ax_(1)^(2)+bx_(1)+c...(i)` and `y_(2)=ax_(2)^(2)+bx_(2)+c....(ii)` `therefore y_(1)-y_(2) = [a(x_(1)+x_(2))(x_(1)-x_(2))+b(x_(1)+x_(2))]` `therefore (y_(2)-y_(1))/(x_(2)-x_(1))=a(x_(1)+x_(2))+b` `therefore "from" (1), a(x_(1)+x_(2))+b=2ax_(3)+b` `(x_(1)+x_(2))/(2)=x_(3)` |
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