   Chapter 2.P, Problem 3P

Chapter
Section
Textbook Problem

# Show that the tangent lines to the parabola y = a x 2 + b x + c at any two points with x-coordinates p and q must intersect at a point whose x-coordinate is halfway between p and q.

To determine

To show: the tangents to parabola y=ax2+bx+c  at any two points with x-coordinate’s p and q must intersect at  point whose x- coordinate is halfway between p and q.

Explanation

1) Concept:

Here we have to use concept of derivative of curve as slope of tangent.

From given x-coordinates of points find y coordinates, where given curve has tangent.

Using given slopes and points we can find equation of tangents.

As tangents intersects each other, equating both equations of tangent we can find x- coordinate of point of intersection.

2) Formula:

i. Slope of tangent

= mtanΑ=dydx

ii. Equation of tangent through a, b is

y-b=mtanΑ (x-a)

3) Given:

Parabola y=ax2+bx+c has tangents at Pp, yp and Q(q, yq).

4) Calculations:

Slope of tangent to given parabola y=ax2+bx+c is given by,

mtanΑ=ddxax2+bx+c=2ax+b

As given parabola has tangents at points Pp,  ap2+bp+c and Q(q,  aq2+bq+c).

Therefore, equation of tangent through point Pp,  ap2+bp+c with slope mtanΑ=2ap+c is given by,

y=2ap+bx-p+ap2+bp+c

Through Q(q,  aq2+bq+c) with slope mtanΑ=2aq+c is given by,

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