If the equation $16x^2 - 24xy + 9y^2 - 8x + 6y - 35 = 0$ represents a pair of straight lines, then the equation of the locus of points equidistant from these two lines is..........

The equation of the circle which passes through the points $(2, 3)$ and $(4, 5)$ and whose centre lies on a straight line $4x - y - 3 = 0$, is

If $\theta$ is the eccentric angle of a point on the ellipse $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$ such that the distance of the point from the center is $5$, then $\theta = $.......

$\lim\limits_{x \to 0}\left[\dfrac{x \cdot \log(1 + 4x)}{\left(e^{4x} - 1\right)^2}\right] = \cdots$