If $$(a,-2 a), a>0$$ is the mid-point of a line segment intercepted between the co-ordinate axes, then the equation of the line is

If the vectors $$\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+m \hat{\mathbf{k}}$$ are coplanar, then $$m=$$

The letters of the word 'LOGARITHM' are arranged at random. The probability that arrangement starts with vowel and end with consonant is

If $$\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4$$, then $$\frac{d y}{d x}=$$