Various Forms Of The Equation Of A Line

The binomial expansion of gives terms of the form: We are given that the coefficients of three consecutive terms , , and form a geometric progression (G.P.). So, the ratio of the coefficients of the terms should be equal: Using the binomial coefficients, this simplifies to: This results in the following equation: By simplifying, we obtain: This simplifies to: Solving this quadratic equation gives no valid values for , so . Next, for the sum of rational terms in the binomial expansion of , the general term is: The rational terms occur when the exponents of the square roots are even, so we consider the terms with even powers of and . We calculate the sum of these rational terms: Thus, .

The differential equation is: Rearranging the terms, we integrate to solve for : Given that , we solve for : Thus, .

We are given the components of along and perpendicular to : where After combining the vectors: Now, we compute :