The chord joining the points and on the curve is parallel to the tangent at a point on the curve. What is the abscissa ( -coordinate) of the point?
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Solution:
As the lines are parallel , their slopes must be equal.
Slope of chord:
x 2 − x 1 y 2 − y 1 = 1 1 − 5 2 2 7 − 5 = 6 2 2 2 = 3 7 ⋯ ( 1 )
Slope of curve= Derivative of the equation of the curve.
y d x d y = 3 x 2 − 1 1 x − 1 5 = 6 x − 1 1 ⋯ ( 2 )
Equating ( 1 ) and ( 2 ) we get,
6 x − 1 1 = 3 7 x = 8
∴ Abscissa or x -coordinate of the point is 8