Question

Write the standard equation for the ellipse with the given conic form.

Answer 1

We complete the square

$4 x^2 + y^2 - 16x + 2y + 13 = 0$

$4 x^2 - 16x + y^2 + 2y = -13$

$4 (x^2 - 4x) + (y^2 + 2y) = -13$

$4 (x^2 - 4x + 4) + (y^2 + 2y +1 ) = -13 +16 + 1$

$4 (x - 2)^2 + (y+1)^2 = 4$

$\dfrac{ (x - 2)^2 }{1^2} + \dfrac{ (y+1)^2}{2^2} = 1$

That's standard form with axes 1 and 2