Find a point on the $y$-axis which is equidistant from the points $A(6, 5)$ and $B(-4, 3)$.

Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are $(0,-1), (2,1)$ and $(0,3)$. Find the ratio of this area to the area of the given triangle.

Find the area of a rhombus if its vertices are $(3,0), (4,5), (-1,4)$ and $(-2,-1)$ taken in order. (in square units)

If $(1, 2), (4, y), (x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order,find $x$ and $y$.

To conduct Sports Day activities,in your rectangular shaped school ground $ABCD,$ lines have been drawn with chalk powder at a distance of $1 \,m$ each. $100$ flower pots have been placed at a distance of $1 \,m$ from each other along $AD,$ as shown in the figure. Niharika runs $\frac{1}{4}$ th the distance $AD$ on the $2^{nd}$ line and posts a green flag. Preet runs $\frac{1}{5}$ th the distance $AD$ on the $8^{th}$ line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags,where should she post her flag?

Find a relation between $x$ and $y$ such that the point $(x, y)$ is equidistant from the points $(7, 1)$ and $(3, 5)$.