#28 Measure you Calibre

Calculus Level 2

The chord joining the points ( 5 , 5 ) (5, 5) and ( 11 , 227 ) (11, 227) on the curve y = 3 x 2 11 x 15 y= 3x^2-11x-15 is parallel to the tangent at a point on the curve. What is the abscissa ( x x -coordinate) of the point?

8 -8 -4 4

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1 solution

Satwik Murarka
Apr 10, 2017

Solution:

As the lines are parallel , their slopes must be equal.

Slope of chord:

y 2 y 1 x 2 x 1 = 227 5 11 5 = 222 6 = 37 ( 1 ) \begin{aligned}\frac{y_2-y_1}{x_2-x_1}&=\frac{227-5}{11-5}\\&=\frac{222}{6}\\ &=37\hspace{2cm} \cdots(1)\end{aligned}

Slope of curve= Derivative of the equation of the curve.

y = 3 x 2 11 x 15 d y d x = 6 x 11 ( 2 ) \begin{aligned}\large y&=3x^{2}-11x-15\\ \frac{dy}{dx}&=6x-11\hspace{2cm}\cdots(2)\end{aligned}

Equating ( 1 ) (1) and ( 2 ) (2) we get,

6 x 11 = 37 x = 8 \begin{aligned}6x-11=37\\ x=8\end{aligned}

\therefore Abscissa or x x -coordinate of the point is 8 \boxed{8}

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