Application of Derivatives

Question

Find a point on the curve y = (x – 2)² at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).

Solution

The equation of curve is
$straight y space equals space left parenthesis straight x minus 2 right parenthesis squared$
$therefore space space space space dy over dx space equals space 2 space left parenthesis straight x minus 2 right parenthesis therefore space space space space slope space of space tangent space equals space 2 left parenthesis straight x minus 2 right parenthesis$
Also slope of line joining (2, 0) and (4, 4) $equals space fraction numerator 4 minus 0 over denominator 4 minus 2 end fraction space equals space 4 over 2 space equals space 2$
$because space space tangent space is space parallel space to space the space line space joining space left parenthesis 2 comma space 0 right parenthesis space and space left parenthesis 4 comma space 4 right parenthesis therefore space space space their space slopes space are space equal therefore space space space space space 2 left parenthesis straight x minus 2 right parenthesis space equals space 2 space space space space space rightwards double arrow space space space space space straight x minus 2 space equals space 1 space space space space rightwards double arrow space space space straight x space equals space 3 When space straight x space equals space 3 comma space space space straight y space equals space left parenthesis 3 minus 2 right parenthesis squared space equals space 1 therefore space space space space space space required space point space is space left parenthesis 3 comma space 1 right parenthesis.$

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