Find the coordinates of the foot of the perpendicular drawn from the point $(0, 5)$ to the line $3x - 4y - 5 = 0$.

Let $ABC$ be a triangle formed by the lines $7x-6y+3=0$,$x+2y-31=0$,and $9x-2y-19=0$. Let the point $(h, k)$ be the image of the centroid of $\Delta ABC$ in the line $3x+6y-53=0$. Then $h^2+k^2+hk$ is equal to

The equation of the line bisecting the line segment joining the points $(a, b)$ and $(a', b')$ at a right angle is

Consider the lines $x(3 \lambda+1)+y(7 \lambda+2)=17 \lambda+5$,where $\lambda$ is a parameter. All these lines pass through a fixed point $P$. One of these lines (say $L$) is farthest from the origin. If the distance of $L$ from the point $(3,6)$ is $d$,then the value of $d^2$ 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

The perpendicular from the origin to a line meets it at the point $(-2, 9)$. Find the equation of the line.