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\mathrm{A\left(x_1, y_1\right)} and \mathrm{B\left(x_2, y_2\right)}  are any two points on the parabola \mathrm{y=a x^2+b x+c}. If \mathrm{P\left(x_3, y_3\right)} be the point on the arc \mathrm{A B} where the tangent is parallel to the chord \mathrm{A B}, then
 

Option: 1

 \mathrm{x_2} is the A.M. between \mathrm{x_1} and \mathrm{x_3}
 


Option: 2

 \mathrm{x_2} is the G.M. between \mathrm{x_1} and \mathrm{x_3}
 

 


Option: 3

\mathrm{x_2} is the H.M. between \mathrm{x_1} and \mathrm{x_3}
 


Option: 4

 none of these


Answers (1)

best_answer

 Slope of tangent at P\left(x_3, y_3\right)=2 a x_3+b=\frac{y_2-y_1}{x_2-x_1} \quad \text { [given] } \ldots \text { (i) }

\mathrm{\left\{\text { As the tangent is } \frac{y+y_3}{2}=a x x_3+b\left(\frac{x+x_3}{2}\right)+c\right\} }
Q     A and B lie on the parabola,

\begin{aligned} &\mathrm{ \therefore y_1=a x_1^2+b x_1+c } \quad \quad \quad \dots(ii)\\ &\mathrm{ \text { and } y_2=a x_2^2+b x_2+c} \quad \quad \quad \dots(iii)\\ &\mathrm{\therefore \quad y_1 y_2=\left[a\left(x_1+x_2\right)\left(x_1-x_2\right)+b\left(x_1-x_2\right)\right]}\\ &\mathrm{\therefore \quad \frac{y_2-y_1}{x_2-x_1}=a\left(x_1+x_2\right)+b}\\ &\mathrm{\therefore from (i), a\left(x_1+x_2\right)+b=2 a x_3+b \Rightarrow \frac{x_1+x_2}{2}=x_3}\\ \end{aligned}
 

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