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

Complete the squares:

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

25(x+1)²+169(y-2)²=3524+25+676=4225.

Divide through by 4225:

(x+1)²/169+(y-2)²/25=1.

(x+1)²/13²+(y-2)²/5²=1.

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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