Find the shortest distance between the point P(2,1) and the graph of y=2x2. Give your answer correct to 3 significant figures

Answer 1

The distance is measure along the line perpendicular to y = 2x² and passing though (2,1)

First find the slope of the tangent line to y = 2x²

y' = 4x

The normal (perpendicular) has a slope of the negative inverse, -1/4

The equation of the normal line is

(y - 1) / (x - 2) = -1/4

4(y - 1) = -(x - 2)

4y - 4 = -x + 2

4y = -x + 6

y = -x/4 + 3/2

Find the intersection of this line and y = 2x²

Then determine the distance between (2, 1) and the point of intersection.