If $P^{\prime}(a, b)$ is the image of the point $P(-1, 2)$ with respect to the line $x-2y+3=0$,then the length of the perpendicular from $P^{\prime}$ onto the line $2x+y-7=0$ is

The unit normal vector to the line joining $i - j$ and $2i + 3j$ and pointing towards the origin is

If $A$ and $B$ are two points on the line $3x + 4y + 15 = 0$ such that $OA = OB = 9$ units,then the area of the triangle $OAB$ is

$y-x=0$ 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

If $A(2, -1)$ and $B(6, 5)$ are two points,the ratio in which the foot of the perpendicular from $P(4, 1)$ to $AB$ divides $AB$ is:

The equation of the straight line which passes through the point $(a \cos^{3} \theta, a \sin^{3} \theta)$ and is perpendicular to $x \sec \theta + y \operatorname{cosec} \theta = a$ is