The ellipse, the hyperbola and a line against them
Find the foci of an ellipse and a hyperbola from the equation, build an ellipse with its centre anywhere, draw the asymptote of a hyperbola and test a line against a parabola.
- سوريا: الصف ١٠
- فلسطين: الصف ١١
- إيران: الصف ١٢
- أفغانستان: الصف ١١
لم يُترجَم هذا المختبر إلى هذه اللغة بعد.
الدليل
An ellipse is the set of points whose distances from two fixed points F and F′ (the foci) add up to ٢a. With the centre at the origin its equation is x٢a٢ + y٢b٢ = ١, with c٢ = a٢ − b٢ and foci (±c,٠). With centre (h,k): (x − h)٢a٢ + (y − k)٢b٢ = ١.
A hyperbola is the set of points whose distances from the two foci differ by ٢a. Its equation is x٢a٢ − y٢b٢ = ١, with c٢ = a٢ + b٢. Careful: in an ellipse c is smaller than a, in a hyperbola larger. Its asymptotes are y = ±ba(x − h) + k.
To test a line against a conic, substitute y from the line into the equation of the conic. Take the discriminant of the quadratic you get: positive, two points of intersection; zero, a tangent; negative, no intersection.