The coordinates of the foot of the perpendicular from $(0,0)$ upon the line $x+y=2$ are

Sum of values of $k$ for which the image of the point $(\lambda^2 + 1, \lambda)$ with respect to the line $y = -3x + 6k$ is the point $(\lambda, \lambda - 1)$ is:

The equations of the perpendicular bisectors of the sides $AB$ and $AC$ of a $\triangle ABC$ are $x-y+5=0$ and $x+2y+5=0$,respectively. If $A$ is $(1, -2)$,then the equation of the straight line $BC$ is

Let $L$ be the line $y = 2x$ in the two-dimensional plane.
Statement $1$: The image of the point $(0, 1)$ in $L$ is the point $\left( \frac{4}{5}, \frac{3}{5} \right)$.
Statement $2$: The points $(0, 1)$ and $\left( \frac{4}{5}, \frac{3}{5} \right)$ lie on opposite sides of the line $L$ and are at equal distance from it.

The point $(2,3)$ is first reflected in the straight line $y=x$ and then translated through a distance of $2$ units along the positive direction of the $X$-axis. The coordinates of the transformed point are

The equation of the line perpendicular to the line $ax + by + c = 0$ and passing through $(a, b)$ is: