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

Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$-axis. Also,find the coordinates of the point of division.

Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by the point $(-1, 6)$.

$A$ median of a triangle divides it into two triangles of equal areas. Verify this result for $\Delta ABC$ whose vertices are $A (4, -6), B (3, -2)$ and $C (5, 2)$. Find the area of each triangle (in square units).

In a classroom,$4$ friends are seated at the points $A, B, C$ and $D$ as shown in the figure. Champa and Chameli walk into the class and after observing for a few minutes Champa asks Chameli,"Don't you think $ABCD$ is a square?" Chameli disagrees. Using the distance formula,find which of them is correct.

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?