Question

By completing the square in X and Y, write an equation for each conic section in standard form. State whether the conic is a circle, elipse, parabola or hyperbola. Then give the center of the conic:

y=3x^2-12x+7

9x^2-54x-4y^2 -16y=-29

Answer 1

For the first equation, completing the square gives:
\[y = 3x^2-12x+7 \Rightarrow y = 3(x^2-4x)+7 \Rightarrow y = 3(x^2-4x+4)+7-12\]
\[\Rightarrow y = 3(x-2)^2-5\]
This is a parabola.

Answer 2

Second equation:
\[9x^2-54x-4y^2-16y = -29 \Rightarrow 9(x^2-6x)-4(y^2+4y) = -29\]
\[\Rightarrow 9(x^2-6x+9)-4(y^2+4y+4) = -29+81-16\]
\[\Rightarrow 9(x-3)^2-4(y+2)^2 = 36 \Rightarrow \frac{(x-3)^2}{4}-\frac{(y+2)^2}{9} = 1\]
This is a hyperbola.