Explanation Given: The perpendicular distances F 1 R 1 and F 2 R 2 Calculation: The standard equation of an ellipse will be given by: ( x a ) 2 + ( y b ) 2 = 1 ... ( 1 ) Equation of the tangent to this ellipse at the point ( x 0 , y 0 ) will be given by: ( x x 0 a ) 2 + ( y y 0 b ) 2 = 1 ... ( 2 ) Now, simplify the equation (2) and we get: ( x x 0 a ) 2 + ( y y 0 b ) 2 = 1 b 2 x 2 x 0 2 + a 2 y 2 y 0 2 − a 2 b 2 = 0 ... ( 3 ) So, the length of perpendiculars on this tangent from the two foci is given by: F 1 R 1 = b 2 x 0 ( − c ) − a 2 b 2 b 4 x 0 2 + a 4 y 0 2 F 2 R 2 = b 2 x 0 ( c ) − a 2 b 2 b 4 x 0 2 + a 4 y 0 2 Now, multiply these perpendicular distances and we get: F 1 R 1 × F 2 R 2 = b 2 x 0 ( − c ) − a 2 b 2 b 4 x 0 2 + a 4 y 0 2 × b 2 x 0 ( c ) − a 2 b 2 b 4 x 0 2 + a 4 y 0 2 F 1 R 1 × F 2 R 2 = b 4 ( a 4 − c 2 x 0 2 ) b 4 x 0 2 + a 4 y 0 2 .....

4th Edition

ISBN: 9781319434526

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Videos

Question

Chapter 12, Problem 55CRE

To determine

That the product of the perpendicular distances $F_{1} R_{1}$ and $F_{2} R_{2}$ from the foci to a tangent line of an ellipse is equal to the square $b^{2}$ of the semi minor axes

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12, Problem 54CRE

Chapter 13.1, Problem 1PQ

Students have asked these similar questions

Show that the Laplace equation in Cartesian coordinates: J²u J²u + = 0 მx2 Jy2 can be reduced to the following form in cylindrical polar coordinates: 湯( ди 1 8²u + Or 7,2 მ)2 = 0.

Find integrating factor

Draw the vertical and horizontal asymptotes. Then plot the intercepts (if any), and plot at least one point on each side of each vertical asymptote.

Chapter 12 Solutions

CALCULUS 4E (LL) W/ ACHIEVE ACCESS

Ch. 12.1 - Prob. 1PQ Ch. 12.1 - Prob. 2PQ Ch. 12.1 - Prob. 3PQ Ch. 12.1 - Prob. 4PQ Ch. 12.1 - Prob. 5PQ Ch. 12.1 - Prob. 6PQ Ch. 12.1 - Prob. 7PQ Ch. 12.1 - Prob. 1E Ch. 12.1 - Prob. 2E Ch. 12.1 - Prob. 3E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

That the product of the perpendicular distances F 1 R 1 and F 2 R 2 from the foci to a tangent line of an ellipse is equal to the square b 2 of the semi minor axes

Solution Summary: The author proves that the product of the perpendicular distances F 1 R 1 and F 2 R 2 from the foci to a tangent line of an ellipse is equal to the square b

Videos

That the product of the perpendicular distances $F_{1} R_{1}$ and $F_{2} R_{2}$ from the foci to a tangent line of an ellipse is equal to the square $b^{2}$ of the semi minor axes

Want to see the full answer?

Chapter 12 Solutions

That the product of the perpendicular distances F 1 R 1 and F 2 R 2 from the foci to a tangent line of an ellipse is equal to the square b 2 of the semi minor axes

Solution Summary: The author proves that the product of the perpendicular distances F 1 R 1 and F 2 R 2 from the foci to a tangent line of an ellipse is equal to the square b

Videos

That the product of the perpendicular distances F1R1 and F2R2 from the foci to a tangent line of an ellipse is equal to the square b2 of the semi minor axes

Want to see the full answer?

Chapter 12 Solutions

That the product of the perpendicular distances $F_{1} R_{1}$ and $F_{2} R_{2}$ from the foci to a tangent line of an ellipse is equal to the square $b^{2}$ of the semi minor axes