Question #224008

By a suitable change of origin, show that the equation 9(x − 5)2 + 16(y + 3)2 = 144 represent an ellipse and that the coordinates of the centre are (5, −3). Find the position vectors of the foci and the ends of the major and minor axes.


1
Expert's answer
2021-08-15T17:35:15-0400

9(x5)2+16(y+3)2=1449(x-5)^2+16(y+3)^2=144


(x5)216+(y+3)29=1\dfrac{(x-5)^2}{16}+\dfrac{(y+3)^2}{9}=1

The equation of ellipse


(xaC)2a2+(yyC)2b2=1\dfrac{(x-a_C)^2}{a^2}+\dfrac{(y-y_C)^2}{b^2}=1

The center C(5,3)C(5, -3)


a=4,b=3a=4, b=3


c=a2b2=4232=7c=\sqrt{a^2-b^2}=\sqrt{4^2-3^2}=\sqrt{7}

Foci: (57,3),(5+7,3).(5-\sqrt{7}, -3), (5+\sqrt{7}, -3).


Ends of the major axis: (1,3),(9,3).(1, -3), (9, -3).


Ends of the minor axis: (5,6),(5,0).(5, -6), (5, 0).



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