Question

Is 4x^2+y^2+16x-6y+9=0 a parabola, a circle, an ellipse, or a hyperbola?

Answer 1

An ellipse with vertical major axis.

Explanation:

`4x^2+y^2+16x-6y+9=0`

Convert to standard form by completing the square:

`4x^2+16x+y^2-6y=-9`

`4(x^2+4x+4)+(y^2-6y+9)=-9+16+9`

`4(x+2)^2+(y-3)^2=16`

`(x+2)^2/4+(y-3)^2/16=1`

Equation of an ellipse with vertical major axis in standard form:
`(x-h)^2/a^2+(y-k)^2/b^2=1`
Where:
`(h, k)` is the coordinate of the center
`2b` is the major axis

So in this case:
Center is at `(-2, 3)`
`a=2, b=4`
Major axis is along the y-axis and the length is: `8`