tangent problem, how a,b,c are in AP

Question

if the tangent to the curve f(x)= x² at any point (c,f(c)) is parallel to the line joining (a,f(a)) and (b,f(b)) on the curve ,then which one of the following is true and how explain it, plzzzzzzzz ?  

1) a,b,c in AP

2) a,b,c in GP

3) a,b,c in HP

Answer 1

Hello Mrutyunjaya,

 

The slope of the curve, and thus the tangent to the curve, at (c,f(c)) is 2c since the slope is f'(x) = 2x ^.

Now the 2 points on the curve are (a,a²) and (b,b²) and the slope of the chord formed by joining these 2 points is   

 

[(b²-a²)/(b-a)] = b+a.

 

Since the lines are parallel the slopes are equal 2c = b+a thus a,c,b are in Arithmetic Progression.

 

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