Question

How do you graph `4x^2+9y^2+24-18+9=0`?

Answer 1

(see below)

Explanation:

Note: I am assuming that the equation should have been:
`color(white)("XXX")4x^2+9y^2+24color(red)(x)-18color(red)(y)+9=0`

The standard form of an ellipse is
`color(white)("XXX")((x-h)^2)/(a^2)+((y-k)^2)/(b^2)=1`

with center at `(h,k)`,
radius on X-axis `=a`, and
radius on Y-axis`=b`

Converting the (assumed) equation into this form:
`color(white)("XXX")4(x^2+6x)+9(y^2-2y)=-9`

`color(white)("XXX")4(x^2+6x+9)+9(y^2-2y+1)=-9+4xx9+9xx1`

`color(white)("XXX")4(x+3)^2+9(y-1)^2=36`

`color(white)("XXX")((x+3)^2)/(3^2)+((y-1)^2)/(2^2)=1`

So we need to draw an ellipse with center at `(-3,1)`,
a radius of `3` parallel to the X-axis, and
a radius of `2` parallel to the Y-axis.