Math help?

Question

Identify the conic represented by the equation
3x^2 + 6xy + 12y^2 + 2x - 5y = -1.

Answer 1

You should 1st let x=x'+h and y = y'+k. Substitute these values for x and y in the given equation. Then collect all the terms involving x' and y'. and set the coefficients of these terms to zero. This will give you two equations in h and k which you can solve simultaneously.

Put the values for h ank k into the translated equation and the new equation will have the x' and y' terms removed, leaving you with just x'^2, x"y', y'^2 and a constant.

We now remove the x'y' term by a rotation of the axes through an angle z , where cot2z=(a-c)/b, where a is the coefficient of x'^2, b is the coefficient of x"y', and c is the coefficient of y'^2.

Solve for sin z and cos z and make the rotation by substituting x' = cos z *x" and y' = sin z y".
This will reduce the equation that contains only x"^2, y"^2 and a constant.

You should now be able to identify the conic.