Find the equation of the straight line passing through $(0,0)$ and the foot of the perpendicular from $(2,4)$ onto the line $x+y-1=0$.

The value of $a$ for which the image of the point $(a, a - 1)$ with respect to the line mirror $3x + y = 6a$ is the point $(a^2 + 1, a)$ is:

If $2x + 3y + 4 = 0$ is the perpendicular bisector of the line segment joining the points $A(1, 2)$ and $B(\alpha, \beta)$,then the value of $13\alpha + 13\beta$ equals $......$

Let $P$ be an interior point of a $\triangle ABC$. Let $Q$ and $R$ be the reflections of $P$ in $AB$ and $AC$,respectively. If $Q, A, R$ are collinear,then $\angle A$ equals (in $^{\circ}$)

What is the reflection of the point $(1, -2)$ with respect to the $y$-axis?

Let $P$ be a point inside a $\triangle ABC$ with $\angle ABC = 90^{\circ}$. Let $P_1$ and $P_2$ be the images of $P$ under reflection in $AB$ and $BC$ respectively. The distance between the circumcenters of $\triangle ABC$ and $\triangle P_1PP_2$ is