# Algebraic Geometry - Bowdoin 1985, Part 1 by Bloch S. (ed.) By Bloch S. (ed.)

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Extra info for Algebraic Geometry - Bowdoin 1985, Part 1

Sample text

Properties of Conics θ 29 θ θ plane mirror θ curved mirror This law applies to all mirrors, no matter whether the reflecting surface is plane or curved. Indeed, in many practical applications the mirror is designed to have a cross-section that is a conic curve – for example, the Lovell radiotelescope at Jodrell Bank in Cheshire, England uses a parabolic reflector to focus parallel radio waves from space onto a receiver. We now investigate the reflection properties of mirrors in the shape of the non-degenerate conics.

Now we can obtain any non-degenerate conic from a conic in standard form by a suitable rotation (x, y) → (x cos θ − y sin θ , x sin θ + y cos θ ) followed by a suitable translation (x, y) → (x − a, y − b). This rotates the axes through an anticlockwise angle θ to align them with the axes of the conic. This moves the centre or vertex of the conic to the origin. Both of these transformations are linear, so that the equation of the conic at each stage is a second degree equation of the type (1); in other words, any non-degenerate conic has an equation of type (1).

Y b F′ –a –b x = –a /e y = (b/a)x a F x = a/e x y = – (b/a)x 1: Conics 18 We summarize the above facts as follows. Hyperbola in Standard Form A hyperbola in standard form has equation x2 y2 − 2 = 1, where b2 = a 2 e2 − 1 , a > 0, e > 1. a2 b It has foci (±ae, 0) and directrices x = ±a/e; and it can be described by the parametric equations x = a sec t, y = b tan t x =a 2t 1 + t2 ,y=b 1 − t2 1 − t2 t ∈ R − {±1}. (t ∈ (−π/2, π/2) ∪ (π/2, 3π/2)). Problem 9 Let P be a point (sec t, √12 tan t), where (t (−π/2, π/2) ∪ (π/2, 3π/2)), on the hyperbola E with equation 2y 2 = 1.