Prove: if b ≠ 0, then the cross product term can be eliminated from the quadratic form ax² + 2bxy + cy² by rotating the coordinate axes through an angle θ that satisfies the equation cot 2θ = a-c/2b.

Astudent is determining the influence of different types of food on the growth rate of spiders. she feeds one group of spiders two caterpillars twice a week. the other group received similarly-sized beetles twice a week. she records the amount of time required for individuals to reach sexual maturity. her hypothesis is that the spiders feeding on caterpillars will have a faster growth rate because caterpillars have a higher protein content compared to beetles. in this experiment, what is the dependent variable? spiders assigned to different feeding groups. time required to reach sexual maturity number of prey items fed twice a week. growth rates of each group.