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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Circle Two circles of radius 4 are tangent to the graph of y 2 = 4 x at the point (1, 2). Find equations of these two circles.

To determine

To calculate: The equations of the two circles passing through the point (1,2) and tangent to the parabola is y2=4x.

Explanation

Given:

The radius of circle is 4. The of equation of parabola, y2=4x. The point (1,2).

Formula used:

Equation of circle, (xx1)2+(yy1)2=r2.

Calculation:

The of equation of parabola,

y2=4x

Differentiate with respect to x,

2yy=4yy=2y=2y

The slope of the parabola at point (1,2) is,

y=22=1

The slope of the tangent line at the point (1,2) is 1. The center of the circle must be lies on the normal line at the point (1,2) on the parabola. From the reciprocal theorem of slope, the slope of the normal line equation is,

m1m2=1(1)m2=1m2=1

The equation of normal line passing through the point on the parabola at (1,2),

y2=1(x1)y=3x

The distance formula,

d=(xx1)2+(yy1)2

The center of the circle 4 units away from (1,2). The distance is,

42=(x1)2+(y2)216=(x1)2+(y2)2

Substitute y=3x in the above equation,

(x1)2+(3x2)2=162(x1)2=16(x1)=±22x=1±22

At x=1+22 the value of y is,

y=3(1+22)=222

The center of the circle is, (1+22,222)

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P-2ESect-3.8 P-3ESect-3.8 P-4ESect-3.8 P-5ESect-3.8 P-6ESect-3.8 P-7ESect-3.8 P-8ESect-3.8 P-9ESect-3.8 P-10ESect-3.8 P-11ESect-3.8 P-12ESect-3.8 P-13ESect-3.8 P-14ESect-3.8 P-15ESect-3.8 P-16ESect-3.8 P-17ESect-3.8 P-18ESect-3.8 P-19ESect-3.8 P-20ESect-3.8 P-21ESect-3.8 P-22ESect-3.8 P-23ESect-3.8 P-24ESect-3.8 P-25ESect-3.8 P-26ESect-3.8 P-27ESect-3.8 P-28ESect-3.8 P-29ESect-3.8 P-30ESect-3.8 P-31ESect-3.8 P-32ESect-3.8 P-33ESect-3.8 P-34ESect-3.8 P-35ESect-3.8 P-36ESect-3.8 P-37ESect-3.8 P-38ESect-3.8 P-39ESect-3.8 P-40ESect-3.8 P-41ESect-3.8 P-42ESect-3.8 P-43ESect-3.8 P-44ESect-3.8 P-45ESect-3.8 P-46ECh-3 P-1RECh-3 P-2RECh-3 P-3RECh-3 P-4RECh-3 P-5RECh-3 P-6RECh-3 P-7RECh-3 P-8RECh-3 P-9RECh-3 P-10RECh-3 P-11RECh-3 P-12RECh-3 P-13RECh-3 P-14RECh-3 P-15RECh-3 P-16RECh-3 P-17RECh-3 P-18RECh-3 P-19RECh-3 P-20RECh-3 P-21RECh-3 P-22RECh-3 P-23RECh-3 P-24RECh-3 P-25RECh-3 P-26RECh-3 P-27RECh-3 P-28RECh-3 P-29RECh-3 P-30RECh-3 P-31RECh-3 P-32RECh-3 P-33RECh-3 P-34RECh-3 P-35RECh-3 P-36RECh-3 P-37RECh-3 P-38RECh-3 P-39RECh-3 P-40RECh-3 P-41RECh-3 P-42RECh-3 P-43RECh-3 P-44RECh-3 P-45RECh-3 P-46RECh-3 P-47RECh-3 P-48RECh-3 P-49RECh-3 P-50RECh-3 P-51RECh-3 P-52RECh-3 P-53RECh-3 P-54RECh-3 P-55RECh-3 P-56RECh-3 P-57RECh-3 P-58RECh-3 P-59RECh-3 P-60RECh-3 P-61RECh-3 P-62RECh-3 P-63RECh-3 P-64RECh-3 P-65RECh-3 P-66RECh-3 P-67RECh-3 P-68RECh-3 P-69RECh-3 P-70RECh-3 P-71RECh-3 P-72RECh-3 P-73RECh-3 P-74RECh-3 P-75RECh-3 P-76RECh-3 P-77RECh-3 P-78RECh-3 P-79RECh-3 P-80RECh-3 P-81RECh-3 P-82RECh-3 P-83RECh-3 P-84RECh-3 P-85RECh-3 P-86RECh-3 P-87RECh-3 P-88RECh-3 P-89RECh-3 P-90RECh-3 P-91RECh-3 P-92RECh-3 P-93RECh-3 P-94RECh-3 P-95RECh-3 P-96RECh-3 P-97RECh-3 P-98RECh-3 P-99RECh-3 P-100RECh-3 P-101RECh-3 P-102RECh-3 P-103RECh-3 P-104RECh-3 P-105RECh-3 P-106RECh-3 P-107RECh-3 P-108RECh-3 P-109RECh-3 P-110RECh-3 P-111RECh-3 P-112RECh-3 P-113RECh-3 P-114RECh-3 P-115RECh-3 P-116RECh-3 P-117RECh-3 P-118RECh-3 P-119RECh-3 P-120RECh-3 P-121RECh-3 P-122RECh-3 P-123RECh-3 P-124RECh-3 P-125RECh-3 P-126RECh-3 P-127RECh-3 P-128RECh-3 P-129RECh-3 P-130RECh-3 P-1PSCh-3 P-2PSCh-3 P-3PSCh-3 P-4PSCh-3 P-5PSCh-3 P-6PSCh-3 P-7PSCh-3 P-8PSCh-3 P-9PSCh-3 P-10PSCh-3 P-11PSCh-3 P-12PSCh-3 P-13PSCh-3 P-14PSCh-3 P-15PSCh-3 P-16PS

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