Question

How do I graph `9x^2+16y^2-36x+32y=92` algebraically?

Answer 1

By grouping `x`'s and `y`'s together,

`Rightarrow (9x^2-36x)+(16y^2+32y)=92`

by factoring out 9 and 16,

`Rightarrow9(x^2-4x)+16(y^2+2y)=92`

by adding and subtracting 4 and 1 in the parentheses,

`Rightarrow9(x^2-4x+4-4)+16(y^2+2y+1-1)=92`

by keeping the first three terms in each group,

`Rightarrow9(x^2-4x+4)-36+16(y^2+2y+1)-16=92`

by completing the squares and adding 36 and 16,

`Rightarrow9(x-2)^2+16(y+1)^2=144`

by dividing by 144,

`Rightarrow(x-2)^2/{16}+(y+1)^2/{9}=1`

Hence, we have an equation of an ellipse

`(x-2)^2/{4^2}+(y+1)^2/{3^2}=1`,

which is 8 units wide, 6 units tall, and centered at `(2,-1)`.

---

I hope that this was helpful.