Find the Orthogonal Trajectories of the Family of Curves $x ^ { 2 } + y ^ { 2 } = k$

question 73

Essay

Find the orthogonal trajectories of the family of curves $x ^ { 2 } + y ^ { 2 } = k$ . Then draw several members of each family on the same coordinate plane.

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Definitions:

Shortgrass Prairie

A grassland ecosystem characterized by the dominance of shortgrass species, adapted to arid conditions and grazing pressure, found in the central United States and other parts of the world.

Temperate Grassland

A grassland characterized by hot summers, cold winters, and less rainfall than is found in a temperate deciduous forest biome.

Mediterranean Sea

A sea connected to the Atlantic Ocean surrounded by the Mediterranean region and almost completely enclosed by land.

Chaparral

A biome with a Mediterranean climate (mild, moist winters and hot, dry summers). Chaparral vegetation is characterized by drought-resistant, small-leaved evergreen shrubs and small trees.