Simplifying by Translating the Axes

What this move does

Translating the axes: the origin moves to $O^{'} (h, k)$ and both axes keep their directions. The new and old coordinates of the same point satisfy

x = x^{'} + h, y = y^{'} + k .

The curve itself does not move; only its equation changes its look. For an equation $A x^{2} + C y^{2} + D x + E y + F = 0$ without an $x y$ -term:

Complete the square: complete the $x$ terms and the $y$ terms separately to perfect squares, and read off $h, k$ .
Translate the axes: put $x^{'} = x - h$ , $y^{'} = y - k$ , and the linear terms vanish (for the parabolic type you can only remove one linear term and the constant).
Look it up: check the signs of $A, C$ . The same sign means elliptic type (an ellipse, a point or no points); opposite signs mean hyperbolic type (a hyperbola or two crossing lines); one of them zero means parabolic type (a parabola, two parallel lines, one double line or no points).

Predict first

To which point should you move the origin to remove both linear terms of $x^{2} + 4 y^{2} - 4 x + 16 y + 4 = 0$ ?

(2, - 2)

(- 2, 2)

(4, - 16)

(- 4, 16)

Choose an answer first; then you can see the result.

Work through one

Simplify 4x² − 9y² + 16x + 18y − 29 = 0

Step 1 / 5

As it is
The new origin is still at $(0, 0)$ and the equation has not changed. $A = 4$ and $C = - 9$ have opposite signs: hyperbolic type.

$4 x^{2} - 9 y^{2} + 16 x + 18 y - 29 = 0$
Complete the square in x
Complete the square in y
The constant term
The standard equation

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: Translating the axes brings it to a standard equation

Without an $x y$ -term, completing the square handles $x$ and $y$ separately, and after moving to the center only $A x^{'2} + C y^{'2} + F^{'} = 0$ is left.

x^{2} - y^{2} - 2 x + 4 y - 7 = 0

: find the center.

Practice

Exercise 1Basic

After translating the axes to remove the linear terms of $x^{2} + y^{2} - 6 x + 2 y - 6 = 0$ , what is the constant term of the new equation?

Exercise 2Basic

Simplify $4 x^{2} + y^{2} - 8 x + 4 y - 8 = 0$ and find the semi-major axis of this ellipse.

Exercise 3Advanced

The focus of the parabola $y^{2} - 4 y - 4 x + 12 = 0$ is $(m, n)$ . Find $m$ .

Exercise 4Challenge

For which $k$ is the graph of $x^{2} - 2 y^{2} + 4 x + 4 y + k = 0$ a pair of crossing lines?

What this move does

Work through one

Simplify 4x² − 9y² + 16x + 18y − 29 = 0

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Getting the signs of the translation formulas backwards