If #A(x_1,y_1)#,#B(x_2,y_2)# are any two points on the parabola #y=ax^2+bx+c# and #C(x_3,y_3)# is a point on the arc AB where the tangent is parallel to the chord AB, then how should it be shown that #x_3=(x_1+x_2)/2#?

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Answer 1 · 2018-03-17T15:48:38

Please refer to the Explanation.

Let, the Parabola be # S : y=f(x)=ax^2+bx+c#.

Given that,

#A(x_1,y_1), B(x_2,y_2) in S :. y_j=ax_j^2+bx_j+c, j=1,2#.

Hence, the slope #m_c# of the chord #AB# is given by,

# m_c=(y_2-y_1)/(x_2-x_1), (x_1!=x_2)#,

#=1/(x_2-x_1){(ax_2^2+bx_2+c)-(ax_1^2+bx_1+c)}#,

#=1/(x_2-x_1){a(x_2^2-x_1^2)+b(x_2-x_1)}#,

#:. m_c={a(x_2+x_1)+b}..............(ast_1)#.

The slope #m_t# of the tangent to #S# at the point #C(x_3,y_3)#

is #[dy/dx]_(C(x_3,y_3))=f'(x_3)#.

#=[d/dx{ax^2+bx+c}]_(C(x_3,y_3))#,

#=[2ax+b]_(C(x_3,y_3))#.

#:. m_t=2ax_3+b..............(ast_2)#.

Knowing that the Chord #AB# and the Tangent at #C# are

parallel, we must have, # m_c=m_t#.

#:.{a(x_2+x_1)+b}=2ax_3+b......[because, (ast_1) & (ast_2)]#.

# rArr x_3=(x_1+x_2)/2#, as desired!

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