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 `(4 - 0)/(4 - 2) = 4/2 = 2`

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

`dy/dx = 2(x - 2)`

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

2(x – 2) = 2

`\implies` x – 2 = 1

`\implies` x = 3

When x = 3, y = (3 – 2)² = 1

Hence, the required point is (3, 1).