How to classify this conic ecwation 4x^2+9y^2-54y-63=0

This looks like an ellipse.

Complete the square for y: 9 (y²-6y-7) = 9 (y²-6y+9-9-7) = 9 ((y-3) ²-16).

Rewrite the equation:

4x²+9 (y-3) ²-144=0; 4x²/144+9 (y-3) ²/144=1; x²/36 + (y-3) ²/16=1.

Standard equation of an ellipse is (x-h) ²/a² + (y-k) ²/b²=1, where (h, k) is the origin or centre.

This is the equation of an ellipse with centre (0,3), a=6 and b=4 (semimajor and semiminor axes).