`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

Options

A. `x_(3)` is the A.M. between `x_(1)` and `x_(2)`
B. `x_(3)` is the G.M. between `x_(1)` and `x_(2)`
C. `x_(3)` is the H.M. between `x_(1)` and `x_(2)`
D. None of these

Answer

A

Explanation

Slope of tangent at P at
`(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)`