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1+e cos where p = a(1 – e?), e is the eccentricity, and a is the semi-major axis of the ellipse. Show that x = r cos o and y = r sin ø satisfy the equation of an ellipse in rectangular coordinates (shifted to the left by c): (x+c)² , y² + = 1, a² 62 where c= ae and b= (a² – c²)1/2 is the semi-minor axis of the ellipse.

1+e cos where p = a(1 – e?), e is the eccentricity, and a is the semi-major axis of the ellipse. Show that x = r cos o and y = r sin ø satisfy the equation of an ellipse in rectangular coordinates (shifted to the left by c): (x+c)² , y² + = 1, a² 62 where c= ae and b= (a² – c²)1/2 is the semi-minor axis of the ellipse.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The equation of an ellipse in polar coordinates (r, Ø) is given by 1+e cos o' where p = a(1 – e2), e is the eccentricity, and a is the semi-major axis of the ellipse. Show that x = r cos o and y = r sin ø satisfy the equation of an ellipse in rectangular coordinates (shifted to the left by c): (x+c)² , y? = 1, a2 62 where c= ae and b = (a2 – c2)1/2 is the semi-minor axis of the ellipse.
Transcribed Image Text:The equation of an ellipse in polar coordinates (r, Ø) is given by 1+e cos o' where p = a(1 – e2), e is the eccentricity, and a is the semi-major axis of the ellipse. Show that x = r cos o and y = r sin ø satisfy the equation of an ellipse in rectangular coordinates (shifted to the left by c): (x+c)² , y? = 1, a2 62 where c= ae and b = (a2 – c2)1/2 is the semi-minor axis of the ellipse.
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