Classify the conic section 25x 2 + 50x + 169y^2 − 676y = 3524. Find the coordinates of its center, vertices, foci, and sketch its graph.
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1 Answer

Complete the squares:

25(x²+2x+1)-25 + 169(y²-4y+4)-676=3524.


Divide through by 4225:



This is the equation of an ellipse with semi-major axis, a=13 and semi-minor axis, b=5.

Centre is (-1,2).

Vertices at (-1±13,2) and (-1,2±5), that is, (12,2), (-14,2), (-1,7), (-1,-3).

Length of focus from centre of ellipse is given by f²=√(a²-b²)=√169-25=√144=12.

Foci at (-1±12,2), that is, (11,2) and (-13,2).



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