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).
The condition for the tangent to be parallel to the chord joining points ( 2 , 0 ) and ( 4 , 4 ) is,
Slope of tangent =Slope of chord (1)
The slope of a line with points ( x 1 , y 1 ) and ( x 2 , y 2 ) is given by,
Slope = y 2 − y 1 x 2 − x 1
Substitute 2 for x 1 , 4 for x 2 , 0 for y 1 and 4 for y 2 .
Slope = 4 − 0 4 − 2 = 4 2 = 2
Hence, slope of the chord is 2 .
The slope of tangent of the curve is given by,
d y d x = 2 ( x − 2 )
Thus, from equation (1),
2 ( x − 2 ) = 2 x − 2 = 1 x = 3
The value of curve y when x = 3 is,
y = ( 3 − 2 ) 2 = 1
Thus, the required point is ( 3 , 1 ) .
The condition for the tangent to be parallel to the chord joining points
The slope of a line with points
Substitute
Hence, slope of the chord is
The slope of tangent of the curve is given by,
Thus, from equation (1),
The value of curve
Thus, the required point is
