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4-. Hyperbolas are pairs of curves that share the same pair of asyrnptotes. A common example is the graph of y = 1fx, which consists of two curves. both asymptotic to the :raxis and the y-axis. Hyperbolas with diagonal asymptotes have equations in one of the Following two forms. x2 2 2 x2 _z_3'_ = 1 1__ = 1 a. b2 b2 o2 Hyperbolas of the rst form open left and right. Those of the second form open up and down. When r1 = b, the asymptotes are perpendicular to each other, with slopes of 1 and 1. a] Solve the differential equation. dr dx \"sill-2 b] Explain why the resulting family of curves represents all possible hyperbola pairs whose asymptotes have slopes of 1 and 1. 5. Given a family of curves. an orthogonal trajectory is a curve that intersects each curve of the family orthogonally [at right angles}. a] Draw a coordinate plane. Draw the family of curves 1? = kyz, where I: is constant b] Find an equation that can be used to reveal the slopes of this family at any point. c] There exists a second family of curves. orthogonal [perpendicular] to the rst family at all crossing points. At any given intersection point. the slope of the curve from the second family must equal the negative reciprocal of the slope of the curve from the rst family. Find a differential equation that can be used to reveal the slopes of the seoond family at any point. d] Solve this to generate an equation for this second family of curves. e] Draw these two families of curves on the same coordinate plane. Your picture should resemble a spider web