BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805
BuyFind

Single Variable Calculus: Concepts...

4th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781337687805

Solutions

Chapter B, Problem 49E
To determine

To calculate: To show that coordinates of P(x,y) satisfy the equation x2a2+y2b2=1 , where c2=a2b2

Expert Solution

Answer to Problem 49E

It is proved that coordinates of P(x,y) satisfy the equation x2a2+y2b2=1 , where c2=a2b2

Explanation of Solution

Given information: An ellipse has a foci (±c,0) and the sum of the distances from point P(x,y) on the ellipse to the foci is 2a

Formula Used:

Distance between two points is given as

  D=(x1+x2)2+(y1+y2)2

Calculation:

Let us consider the below figure

  Single Variable Calculus: Concepts and Contexts, Enhanced Edition, Chapter B, Problem 49E

From above figure,

  F1P=(x+c)2+y2F2P=(xc)2+y2

Also, given that thesum of the distances from point P(x,y) on the ellipse to the foci is 2a

Thus,

  F1P+F2P=2a(x+c)2+y2+(xc)2+y2=2a(x+c)2+y2=2a(xc)2+y2

Squaring both sides of equation,

  ((x+c)2+y2)2=(2a(xc)2+y2)2(x+c)2+y2=(2a)2+(xc)2+y24a(xc)2+y2x2+c2+2xc+y2=4a2+x2+c22xc+y24a(xc)2+y24xc=4a24a(xc)2+y2xc=a2a(xc)2+y2xca2=a(xc)2+y2

Again, squaring both sides of equation

  (xca2)2=(a(xc)2+y2)2c2x2+a42a2cx=a2((xc)2+y2)c2x2+a42a2cx=a2(x2+c22cx+y2)c2x2+a42a2cx=a2x2+a2c22a2cx+a2y2c2x2+a4=a2x2+a2c2+a2y2c2x2a2x2=a2c2a4+a2y2(c2a2)x2=(c2a2)a2+a2y2b2x2=b2a2+a2y2b2x2+a2y2=b2a2x2a2+y2b2=1

Conclusion:

Hence, coordinates of P(x,y) satisfy the equation x2a2+y2b2=1 , where c2=a2b2

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