- Calculus, Volume 1: One Variable Calculus, with an Introduction to Linear Algebra
- Tom M. Apostol
- Second Edition
- 1967
- 978-1-119-49673-1
13.25. Miscellaneous review exercises on conic sections
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(a) Prove that the collection of all parabolas is invariant under a similarity transformation. That is, a similarity transformation carries a parabola into a parabola.
(b) Find all the parabolas similar to $y = x^2.$ - (a) What happens to the focus and directrix line after the parabola undergoes a similarity transformation?
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(a) Let $X = (x, y)$ be a point on the parabola with the standard form equation $$y^2 = 4cx$$Then, let $tX = (tx, ty)$ be the similarity transformation of $X,$ where $t \neq 0.$ The above equation becomes $$y^2 = 4kx$$ where $k = \frac{c}{t}.$ But this describes a parabola with focus at $(k, 0)$ and directrix line $x = -k \quad \blacksquare$
(b) Since we know that all parabolas are invariant under similarity transform, then for any $C \neq 0,$$$y = 4Cx^2$$ is similar to $y = x^2 \quad \blacksquare$