Translate to (−1,0) and rotate by bisectors

The new axes pass through the point (−1, 0) and are parallel to the lines bisecting the angles between the original axes, i.e. the lines y = x and y = −x. So the new origin is (−1, 0) and the new axes are rotated by 45°.

Transformation formula

For a shift of the origin to (h, k) = (−1, 0) followed by a rotation through θ = 45°:

x = −1 + (X − Y)/√2 y = 0 + (X + Y)/√2

Substitute

The given equation is

(x − y)² + x − 3y = 0

Now

• x − y = −1 + (X − Y)/√2 − (X + Y)/√2 = −1 − √2 Y
• (x − y)² = 1 + 2√2 Y + 2Y²
• x − 3y = −1 + (X − Y)/√2 − 3(X + Y)/√2 = −1 − √2 X − 2√2 Y

Adding:

1 + 2√2 Y + 2Y² − 1 − √2 X − 2√2 Y = 2Y² − √2 X

The transformed equation is 2Y² − √2 X = 0, i.e. Y² = X/√2.

This is a parabola with its vertex at the new origin (−1, 0) and axis along the new X-axis (the line y = x + 1, through (−1, 0) with slope 1), latus rectum 1/√2.