The distance between the point $(2, 1)$ and the image of the point $(3, -1)$ with respect to the line $2x + y - 1 = 0$ is

If $2x + 3y = 5$ is the perpendicular bisector of the line segment joining the points $A\left(1, \frac{1}{3}\right)$ and $B$,then $B$ is equal to

$A$ ray of light is incident along a line which meets another line,$7x - y + 1 = 0$,at the point $(0, 1)$. The ray is then reflected from this point along the line,$y + 2x = 1$. Then the equation of the line of incidence of the ray of light is

Let $A \left(\frac{3}{\sqrt{a}}, \sqrt{a}\right)$ with $a > 0$ be a fixed point in the $xy$-plane. The image of $A$ in the $y$-axis is $B$,and the image of $B$ in the $x$-axis is $C$. If $D(3 \cos \theta, a \sin \theta)$ is a point in the fourth quadrant such that the maximum area of $\triangle ACD$ is $12$ square units,then $a$ is equal to:

The image of the family of lines $(\lambda + 2)x + (\lambda - 1)y - (8\lambda + 1) = 0$ in the line mirror $y = x$ is (where $\lambda$ and $\mu$ are parameters):

$y-x=0$ is the equation of a side of a triangle $ABC$. The orthocentre and circumcentre of the triangle $ABC$ are respectively $(5,8)$ and $(2,3)$. The reflection of the orthocentre with respect to any side of the triangle lies on its circumcircle. Then the radius of the circumcircle of the triangle is