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A family of ellipse whose one member is shown in the figure is not expected to be self-orthogonal. But, when the equation is analysed using differential equation technique, one finds that the equation does represent self-orthogonal trajectory. How does one visualise this type of trajectory?
The answer lies in the possible values of $\lambda$ which is a parameter or arbitrary constant. The expression can become negative for several negative values of $\lambda$ and hence the curve can become hyperbola. Now, one can visualise self-orthogonality in the following manner:
### Sum of the coefficients in the expansion of $(x+y)^n$ ....
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$ $\therefore 2^n = 4096 =2^{12}$ $\Rightarrow n = 12$ Greatest coefficient = ${}^{12}{C_6} = 924$