Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
You are not logged in.
Post a reply
Topic review (newest first)
In questions of this sort I tend look for a transformation to get rid of the term:
When this is substituted into the original equation, the term in is
We want this to vanish, so any such that and will do. So we take . Hence
Thus under the transformation the curve becomes the hyperbola . Furthermore as the transformation
represents a clockwise rotation of 45° about the origin followed by an enlargement of at the origin, the conic section is preserved, i.e. the original curve is indeed a hyperbola.
NB: Be careful when using linear transformations on curves: only rotations, reflections and enlargements/contractions by a nonzero factor preserve conic sections. Any other transformation may distort the curve and alter its original nature.
So we now have a more familiar XY = 0 (the rectangular hyperbola)
Now substitute* X = x/a - y/b and Y = x/a + y/b
* substitutions like these preserve the hyperbolic nature of the curve.
and then replacing x1 by x1+ √2.
You might download this
rename the file to Rotation of Axes.pdf This will explain some of this, won't make you as good as scientia or bob bundy with these transformation problems but it is a start.