The 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) is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (3, 1)
Parallel lines have equal slopes. The slope of the chord is
\( \displaystyle m = \frac{4 - 0}{4 - 2} = 2 \)
The slope of the tangent is the derivative:
\( \displaystyle \frac{dy}{dx} = 2(x - 2) \)
\( 2(x - 2) = 2,\quad x = 3 \)
\( y = (3 - 2)^2 = 1 \)
So the point is (3, 1), which lies between x = 2 and x = 4, as the mean value theorem guarantees.