Prove that the collection of all parabolas is invariant under similarity transforms.
Find all parabolas similar to the parabola .
Incomplete.
Related
One comment
S says:
a) Take any parabola. Then fit the coordinate system such that the x axis is the parabola axis, and y is parallel to the directrix. Then we have cartesian equation of the parabola to be (y-y_0) = k(x-x_0)^2. If we replace y=ty and x=tx and t 0, we get (x-x_1)^2=k_1(y-y_1), which is again a parabola.
b) Replacing y=ty and x=tx we get x^2=y/k=Cy.
Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment): Cancel reply
.
a) Take any parabola. Then fit the coordinate system such that the x axis is the parabola axis, and y is parallel to the directrix. Then we have cartesian equation of the parabola to be (y-y_0) = k(x-x_0)^2. If we replace y=ty and x=tx and t 0, we get (x-x_1)^2=k_1(y-y_1), which is again a parabola.
b) Replacing y=ty and x=tx we get x^2=y/k=Cy.