Question

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

Answer 1

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).