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The standard form for conic sections includes a parabola, circle, ellipse, and hyperbola. The equation for a parabola is y - k squared = x - h 4p. For a circle, the equation is x squared + y squared = r squared. An ellipse's equation, specifically a horizontal one, is x - h over a squared plus y - k over b squared = 1. Lastly, a vertical hyperbola is represented by x - h squared over a squared - y - k squared over b squared.