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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Using a ParabolaConsider the parabola

x 2 = 4 y

and the focal chord

y = 3 4 x + 1.

(a) Sketch the graph of the parabola and the focal chord.

(b) Show that the tangent lines to the parabola at the endpoints of the focal chord intersect at right angles.

(c) Show that the tangent lines to the parabola at the endpoints of the focal chord intersect on the directrix of the parabola.

(a)

To determine

To graph: The parabola x2=4y and also graph the focal chord y=34x+1 in the same graph.

Explanation

Given:

The equation for parabola is:

x2=4y

The equation for focal chord is:

y=34x+1

Graph:

Rewrite the equation of parabola and the focal chord as function of y varying with respect to the values x,

y=x24y=3x4+1

In order to draw the graph, consider the different values of x and use the above formulas to calculate the respective values of y.

Table-1 below shown different values of x and y for parabola:

xy=x2469442100

(b)

To determine

To prove: The tangent lines to the parabola x2=4y at the endpoints of the focal chord intersect at right angles.

(c)

To determine

To prove: The tangent lines to the parabola x2=4y at the endpoints of the focal chord intersect on the directrix of the parabola.

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