If the line $y = mx + 1$ meets the circle $x^2 + y^2 + 3x = 0$ in two points equidistant from and on opposite sides of the $x$-axis,then

If a circle passes through the points of intersection of the lines $\lambda x - y + 1 = 0$ and $x - 2y + 3 = 0$ with the coordinate axes,then the value of $\lambda$ is:

If $(\frac{1}{10}, \frac{-1}{5})$ is the inverse point of a point $(-1, 2)$ with respect to the circle $x^2 + y^2 - 2x + 4y + c = 0$,then $c =$

Let $\theta$ be the angle between the circles $S \equiv x^2+y^2+2x-2y+c=0$ and $S' \equiv x^2+y^2-6x-8y+9=0$. If $c$ is an integer and $\cos \theta = \frac{5}{16}$,then the radius of the circle $S=0$ is

Find a possible equation of a chord of the circle $x^2 + y^2 = 100$ which passes through the point $(1, 7)$ and makes an angle of $\frac{2\pi}{3}$ with the origin.

$A$ person $X$ is running around a circular track,completing one round every $40 \ s$. Another person $Y$ running in the opposite direction meets $X$ every $15 \ s$. The time,expressed in seconds,taken by $Y$ to complete one round is