To determine To calculate: The location of foci on an elliptical billiard table if the length of table is 8 feet and the width of table is 5 feet . Explanation Given information: The provided length and width of the elliptical billiard table is 8 feet and 5 feet respectively. Formula used: This figure shows the focus points F 1 and F 2 , Consider the figures where d 1 and d 2 are the distance of the point P from F 1 and F 2 respectively. The points F 1 and F 2 are called foci (‘focus’ for single). These points F 1 and F 2 are fixed for ellipse. The sum of distances d 1 and d 2 from point P on the locus of ellipse will always be constant. The distance d between any two points ( x 1 , y 1 ) a n d ( x 2 , y 2 ) is given by the distance formula as: d = ( x 2 − x 1 ) + ( y 2 − y 1 ) . Calculation: Let the foci of the ellipse are at ( c , 0 ) and ( − c , 0 ) , where these points are on the major axis of the ellipse. Now the length and width of the table is 8 feet and 5 feet . Consider the standard ellipse equation with center at ( 0 , 0 ) , x 2 a 2 + y 2 b 2 = 1 , where a and b are x and y intercepts respectively. Therefore by comparison, a = 8 2 = 4 And b = 5 2 = 2.5 Now write the equation of ellipse in standard form, x 2 4 2 + y 2 2.5 2 = 1 Therefore the points on the x − axis are ( 4 , 0 ) , ( − 4 , 0 ) and on y − axis are ( 0 , 2.5 ) and ( 0 , − 2.5 ) . Now to find the foci of the ellipse use the concept of foci and ellipse which means the sum of distances d 1 and d 2 from point P on the locus of ellipse will always be constant. Let d 1 and d 2 are the distances of the point ( 4 , 0 ) from ( c , 0 ) and ( − c , 0 )

Intermediate Algebra For College Students (10th Edition)

10th Edition

ISBN: 9780134758992

Author: Allen R. Angel, Dennis Runde

Publisher: PEARSON

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is…

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Chapter 10.2, Problem 54ES

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To calculate: The location of foci on an elliptical billiard table if the length of table is 8 feet and the width of table is 5 feet.

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Chapter 10.2, Problem 53ES

Chapter 10.2, Problem 55ES

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Intermediate Algebra For College Students (10th Edition)

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