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

$\text{\bf{We know that if a tangent is parallel}}\\\text{\bf{ to the chord joining the points (2,0) and (4,4),}}\\\text{\bf{ then Slope of the tangent = slope of the curve}}$

now,slope of chord joining the points (2,0) and (4,4) is $\frac{4-0}{4-2}=2$

now, slope of tangent of curve = dy/dx
so, differentiate y with respect to x
$\frac{dy}{dx}=2(x-2)^{2-1}=2(x-2)$

hence, slope of tangent = 2(x - 2)
now, slope of tangent = slope of chord
2(x - 2) = 2
=> x - 2 = 1
=> x = 3 so, y = (3 - 2)² = 1
hence, point on the curve at which the tangent is parallel to the chord is (3, 1)