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).

Answer 1

If a tangent is parallel to the chord joining the points (2, 0) and (4, 4), then the slope of the tangent = the slope of the chord.

The slope of the chord is \(\frac{4-0}{4-2}\) =\(\frac42\)=2.

Now, the slope of the tangent to the given curve at a point (x, y) is given by,

\(\frac{dy}{dx}\)=2(x-2)

Since the slope of the tangent = slope of the chord, we have:

2(x-2) = 2

x-2=1=x=3

when x=3,y=(3-2)²=1

Hence, the required point is (3, 1)